The OP's problem was supposedly that his wife would not accept cut expenses or working, and, according to him, wanted a situation with negative cashflow. Forget FIRE entirely. Negative cashflow.
It is possible the situation was embellished, it's possible the wife merely said "SAH or nothing" and would not thereafter listen to the conversation, it's possible she had a post-partum condition, we have no idea. If, however, she wanted to spend even more money and not work and also stay at home, consequences be damned, it's not monstrous that the OP wanted to discuss the finances, too. The bank will not care for your good reasons.
A lot of goal-oriented guys need to develop a much better delivery on topics where they undervalue the emotional content involved for the other person. No, emotions and rationality are not separate, but emotional content certainly has a higher or lower impact and priority for different people in different situations. Good luck convincing a ravenously starving person to settle down and think carefully. There are leaders who ask about the goal first, and there are leaders who ask about the team first. With a partner, you are always trying to address both enough to reach success and happiness. Compatibility is about communicating even when those priorities don't obviously match, and he's literally asking how to have a discussion. I could understand giving him shit over bias, but not thereafter saying "you should work together" as if he weren't here asking right now how to have a discussion!
What the OP realized is very important. He may be arrogant about it or not - I think we read too much into "rationally discussing" in the thread title, frankly. I think all the title proves is that the OP values rationality. The fact is, he doesn't know how to have that conversation. OP, the recommendations are out there, and I'll try to give you some, because there are entire systems devoted to how to approach those difficult conversations to get better results for both parties. If rationality is your thing, embrace the system!
The sorts of things I'm describing may sound like bullshit to a goal-oriented person who is less impacted by emotion, but you have to overcome that bias, if you have it, to help your wife, and to help your team. Get past your own frustration. Watch "It's Not About the Nail" on Youtube, laugh, acknowledge that others understand your frustration, then MOVE ON and focus on how to heal your partnership, because focusing on your frustration won't do that.
The principles of negotiating start with asking what the other wants, working in good faith to understand the other's wants and needs, asking questions and reflecting so you can help heal emotional panic (which is often built around the fear that the emotional topic won't be addressed satisfactorily), and sharing your own wants and needs as well (that knowledge will hopefully reduce your concern that you won't be heard). Not every marriage can be saved, but many more than are can be.
The limit of therapy is that it's not on your mind enough, and a lot of partners are too likely to hear "listen!" Then, they try listening once because they're following a script, not being empathetic enough, then say "no I tried that and it didn't work." Learn about communication (it's a lot more complex than "talk a lot"). Go to therapy and ask for follow-up books. Tell your wife about techniques you learned to try on yourselves to solve your problems and make you stronger. Therapy as a team translates directly into teamwork if both parties let it.
It may be that your concern that your needs won't be met is hurting your ability to reach out to your wife successfully. You have to prioritize reaching out to her first. Emotional panic cannot be talked through as long as the cause is still there. You wouldn't chat about someone's 401k while they were dangling off a roof. Consider the emotional content of separating from the child to be, for your wife, of similar gravity, whether you experience that gravity or not. She may expect or need "JESUS honey are you ok?" (and she may need a lot of time). If you respond instead "Yes. /sigh. Yes, yes, I REMEMBER, you told me already that you're hanging from the roof, I do listen, but if we can just talk about our finances for a minute...".
Anxiety and fear, like hunger, are not easy to negotiate with. If you can't reach your wife through what you consider totally irrational behavior, the first step is probably HEARING why she is so emotional. Terror that you won't be heard is an easy way to begin ignoring your partnership. Are you doing this? You are already afraid she is doing it to you, so you may well have picked up dismissive behaviors yourself. You can't have a conversation where your objective is to get to what you want, even if what you want is totally rational and important, and learning to do that is hard. It won't matter who was right if you're divorced.
1. Start with concern about her. Talk in whatever setting would be least likely to cause additional stress. By request, I have even started conversations by email for someone with high anxiety, which I initially was totally against, but they helped the person significantly. We would pick them up in person when the emotional impact that something much more terrible must be around the corner had faded. Stay focused and don't react or fight - the more flippant or anxious she is, the less she feels your concern, which means the more she needs to feel it and the more of it you need to show right then.
2. Listening doesn't mean enduring whatever is said. Don't defend yourself. Reflect by summarizing what she says and asking if you have it right. I have solved many fights by saying "You think because of x and y, my first motive was z?" Sometimes she says "I don't know," or something in anger, or even laughs and says she "guesses that can't be right." Hug. Pause. Tell her you care about her, you don't want her to feel bad, and you wish you had known the effect that behavior would have had if it wasn't your intent. The number one cause of failed communication is that we stop listening, and focus only on our agenda, or that we fear we won't be heard, and that fear can be powerful and dangerous. You can get halfway through listening and disagree with an appraisal of your behavior, then ruin the whole thing by defending yourself. Instead of hearing her, you switched to telling her she's wrong. If a beloved aunt/grandmother/best friend/whatever said you were being selfish and that would stun you long enough to make you reflect, consider that you should do the same if somehow your wife got (whatever appraisal of your behavior) that she did.
3. Take your time. If your attitude is that nothing is more important than your partnership, this will automatically turn into better results because it will be easier for her to be on board when she knows she will be heard. Whether reasonable or not, if she thinks it's finances or else, kid or bankruptcy, you win or I win, everyone loses. It may be a huge mistake to assume she doesn't care about the finances. She may need to be heard and then find her own time to come around. A successful relationship requires you to assume this, though that may be scary. It takes a lot of trust. There is no 100% chance that she will. But there's a lot you can do to help, by becoming more communication-fit yourself, and offering as much trust as you can.
The book Difficult Conversations (Stone, Patton & Heen) of the Harvard Negotiation Project is one of a handful that stuck with me. It's an entire book of difficult habits to learn that take faith and patience but tend to automatically spread to people close to you once they feel you're hearing them and reaching out. I think the classes I took on negotiation were the most helpful of my entire academic career. We are all in situations where we feel we know something well, and someone else doesn't. In partnerships, partners need to know they have our trust before anything else, even if they're asking to do something we think is "obviously" foolish, because the bigger thing for their growth or well-being is not the task, it's having our trust. I can't say tanking your finances is a wise choice, but I can say that some people need to know certain things have priority. Your wife may need to know that not separating from her child has priority if she's terrified that your goals will demand separation. Yes, she may even later decide to when it's not so emotionally fraught. From what I've heard from others posting, the powerful aversion to leaving the child anywhere ever is not likely to last forever.
Anecdote on trust: I was with my SO, and I was very exasperated that she wouldn't let me help with a college class. She was going to fail. I had all the answers. I had taken the class. I knew it inside and out. She didn't want help. I said ok and kissed her and went on my way. She failed and later left college. Her parents were doom and warnings. I said I had faith in her. She told me a few times that knowing she was trusted before failing, and after failing, ultimately proved far more important than the college "task" she was going through but never cared much about in the first place. She tried many new things she was once afraid to because her confidence grew as her ability to trust grew, from being trusted herself. She's now a freelance artist and writer running her own business, does odd part-time jobs, still has anxiety, but long since recovered from depression and dystemia.