OK, so I want to start by saying that I feel your pain, and that I work in the same area and so get that errors matter. Even errors of the overprotective kind -- you don't want reports saying action is necessary when it's not, because that creates the impression that you should have done something and were negligent by not doing so.*
But the reality is that you have this coworker, until management figures out she's a doofus, and you need to figure out how to be the good guy here, or else it's just going to be a he-said, she-said, which is when the person who is liked more is kept and the one who is liked less is let go.
So I'm going to be pretty direct here: you are a big part of the problem. You clearly know this stuff better than she does -- I have zero doubt that you're better at your job than she is and the company really needs you. But the way you engage with her is counterproductive, off-putting, and -- as you've seen -- undercutting your own reputation. So the first thing is to please understand that your job isn't to be the person with the right answers -- it's to be effective in developing and implementing an EHS program. And that means figuring out how to change your communication style to bring out her strengths, not force her to double-down and defend her mistakes. You need to realize that any situation that ends in an "I'm right and you're wrong" is a failure -- your failure, even if you're substantively right.
No manager ever gets people who are 100% right all the time. If she was as good at the job as you are, she'd have gotten your job instead of hers. So don't jump on her for being wrong -- figure out how to work with her strengths and help her learn the rest.
I'm going to digress into a little story here, because I think it is instructive. There is a particular male-female communication pattern that I have seen, particularly with detail-oriented men, and it is very destructive. I have a guy who works for me who was negotiating a big settlement. He was making a lot of progress on the issues -- and he still almost got us fired (as in, I had to fly across the country at the drop of a hat to both insert myself into the negotiations and salvage the client). Why? Because our client was female. And she would express her concerns, and he would understand that there was no way we'd get what she wanted, and so he'd basically tell her she was wrong, blow off her points, and move on to the next bit. He was so focused on the details of the settlement language that he was unable to step back and see the big picture: that there was a business reason why she proposed what she did, and maybe there was another way to get there even if her proposed language was a nonstarter (that's what I figured out on my visit). But what was driving her nuts was the impression he left that he thought she was stupid, and even moreso that he was not hearing her -- that she was talking to a brick wall.
And I had the exact same experience with him: I called him up to talk to him about the case and to tell him the client was upset, and I'd say the client wasn't happy, and he'd say well of course they were happy, look at the progress we'd made. I had literally just gotten off the phone with the client who said she was about to get other counsel, and he's insisting that I am wrong about what my client just directly told me. I had to basically yell at him and quote what the client had just said to get him to understand that there was an issue. At that point, I would have fired him if I could have, I was so frustrated! Even though he was actually doing a good job on the substance.
How does that relate to you? Because you can be right and still be wrong. When you jump in to correct every little error, you are asserting your dominance/power, and it is belittling to the person you are talking to. That is true whether it is male or female, but women get this a lot, and so generally every woman who has a backbone is going to bristle and double down in response. This is counterproductive, even when you are 100% right on the substance.
What you need to do is come up with a generic response that makes her feel heard, that validates her concerns, and that then redirects her in the appropriate direction. In other words, look for something that she's doing right, build on that, and then set it up so she figures out the right answer -- in private, on her own, so it can then be her idea. So for ex, how's this for the shower issue:
Her: We should do a temporary shower.
You: [think: boy that's stupid, we need permanent ones - but recognize attempt to be helpful, even if it's misguided. Then say]: You know, that might be a really good temporary solution. I think the regulations require permanent showers so someone can get there within a few seconds, but that is probably going to take time and money to get installed. So while we're looking into that, can you see what our options might be for a temporary shower?
Her: No, a temporary one is all we need, we should just do that and save all the money.
You: [think: grrrrrrrrrr. Say:] Hmm, that's not my impression -- I thought I remembered that the regulations require something you can get to within a few seconds, and I don't think setting up a temporary shower will meet those requirements. But can you look into that? I'd sure like to save the money if OSHA will be happy with it.
[repeat "can you confirm that?" and "again, I'd love to if we can do it, so can you confirm that for me?" or "look, Management is going to make me demonstrate that this will bring us into compliance, so can you pull together the documentation to get that through?" as many times as necessary]
Note that you are validating her attempts to be helpful and her creative thinking, and then redirecting her to do the work to justify it. The only extra "cost" to you is biting your tongue and stamping down your frustration. You don't need to lecture her about how a pump works, or the laws of physics, or anything else, because boy is that condescending. All you need to do is show some sort of interest in her attempts to be helpful, note that you have a question about whether that's feasible, and then put the burden on her to investigate and figure out if it will work.
And when you do need to correct her, there are ways to do it without saying "you're wrong." Like the MSDS/MDSS/SDS bit -- it's reasonable to want employees not to be confused, but you can accomplish that goal by saying "you might also hear them referred to as 'SDS,' because OSHA keeps changing the terminology." Telling them that the "real" term is "SDS" and that "MSDS" is wrong and outdated is unnecessary to avoid confusion, isn't it? The only reason to add that in is to emphasize that you are right and she is wrong. And that publicly undermines her and makes you look petty to everyone in the room, because it tells everyone that you need to be right above all else, even if it means throwing your coworkers under the bus.
I think the other part you're missing is the triage bit. If you guys have such a crappy program, you've got a ton of work to do to get into compliance. So don't let the perfect be the enemy of the good. So what if you let her move forward with a temporary shower -- isn't that better than no shower at all? If you get an OSHA inspection, would you rather say "well, we've been investigating and designing permanent showers and that'll be another year before we get them installed," or would you rather say "well, we figured out it will take a full year to get permanent showers designed and installed, so we've put that contract out, but in the interim we've brought in a temporary shower to provide what protection we can"?
In short, she is clearly trying -- she's busy, she's interested in the area, she is trying to help you out, and she is trying to build a name and reputation in the company. Same as you are. If you find ways to encourage her efforts -- even when she's wrong in some way -- you can turn her into an ally and serve as a really valuable mentor. Yes, it takes a little more care in your conversations, a little more effort on your part. But how much time are you wasting now explaining the laws of physics and being frustrated and angry with her? She is what she is, she knows what she knows. So rather than wish she was someone else, figure out how to make the best of what she has to offer, and give her the opportunity to flesh out her own knowledge.
Final anecdote: when I was a baby lawyer, I did an assignment for the guy who hired me -- one of the two smartest people I've ever met in my life, a guy who has been doing this stuff since I was in elementary school, a guy who knew more in his little finger than I'd ever hope to know, etc. I gave him an answer, he passed it along to the client. The next day, I discovered I was wrong -- no question, 100% wrong, I had missed an exception to the exception. I told him. We called the client together, and he told the client that he had gotten it wrong, that he had missed the exception, and then explained how we'd fix the problem. I was flabbergasted -- I totally deserved to be hung out to dry with the client, but not only did he avoid blaming me, he threw himself under the bus.
I have now worked for him for over 20 years, and I don't even return headhunters' calls. Because he was loyal to me -- he understood that I had made an honest mistake, he did not expect me to know everything, and he gave me a chance to fix it while still protecting my reputation and client relationships. And there's no way in hell I'd give that up for any amount of money. That one action bought years of loyalty and dedication. He gave me a chance to be imperfect and didn't hold it against me, because he recognized the other good qualities I brought to the job. OTOH, the job I had where my boss micromanaged me, second-guessed everything I said, and treated me like I was incompetent? I got the hell out of there as soon as I could.
So: which guy do you want to be?
*I actually worked on a criminal case where a very well-intentioned employee wrote a bunch of letters saying that the company needed to do X, and was the government's star witness at trial. And the company was convicted -- even though when EPA developed the rule, they expressly said you do NOT need to do X. The employee didn't know any better, no one took the time to explain anything to her, and she was extremely persuasive on the stand. And that's all it took.