It's legit to not want to pump or use formula, but she needs to be prepared to deal with the repercussions of it. It sounds like she's ok being woken up, so maybe it's just that you need to accept that it's ok to do that?
This.
First, congratulations!!
Second, from someone who's been there: the first few weeks/months as a first-time mom are among the most intense and powerful of your life; the combination of overpowering hormones, overwhelming responsibility, and zero sleep can make for a ton of self-doubt and pigheaded insistence on stupid things. (Ask me how I know)
So your primary job at this point is to follow her lead and show that you trust her decisions. I truly cannot put into words how powerful some of the breastfeeding messages and beliefs are; it is extremely easy to judge yourself as a success/failure as a mom based on meeting some arbitrary version of "best." (Again, ask me how I know!). So what Captain Fire said is exactly right: your wife's priority is to breastfeed exclusively and not pump. The consequence that comes along with that decision is that she gets to do all the night feedings. You deciding that she needs to pump so you can help is suggesting that her decision is the wrong one -- that is about your need to help more than it is about her needs.
At the same time, it would be completely unreasonable for her to demand that you hold a screaming, hungry baby for hours so she can sleep more, while not giving you the tools necessary to feed the baby. So do what you can to soothe the baby for a while, and then when it's clear that it's hunger, bring the baby in to feed, stay awake so your DW doesn't have to, and then take the baby away after. And if that doesn't allow her to get enough sleep, let her know that you are willing to support her in however she (reasonably) wants to fix it -- i.e., if she wants to suck it up and pump, or supplement with formula, you will take over that midnight feeding on your own and leave her in peace.
FWIW, my DS was
always hungry -- he grew like a weed, maxed out the growth charts, and still couldn't get enough food. Turns out he just had a really, really,
really strong need to suck; we could have avoided a lot of breastfeeding wake-ups if I had been willing to try a bottle of water, just so he could satisfy that craving (he liked the pacifier too but wasn't fooled -- he could tell he wasn't getting anything!). So try those other things -- pacifier, bottle of warm water, lots of strong shushing and tight swaddling and all that stuff -- but again, if they don't work, just feed the baby.
Finally, for you: do not worry about trying to share the baby burden equally right at this specific moment in time. When your wife wants to breastfeed exclusively, you simply cannot. So let her take the lead on that right now, and focus on taking the baby when you can (like bath time, or designated daddy-baby times during the day/on weekends that get longer as the baby can go longer between feedings) and supporting her in other things when you can't (groceries, cooking, cleaning, etc.). And trust that you have a long, long time to equal out that burden. "Fair share" is measured over years, not minutes.