I'm really sorry to hear about your father-in-law.
Obviously, time is short, but there are two "trunk" books to be read when you're able: Difficult Conversations by Douglas Stone and Getting to Yes by Roger Fisher and William Ury, both of the Harvard Negotiation School. They are the gold standard and most other books on the topic are pale imitations. I used to believe that I was great at communication and bad at negotiation, but the opposite was true; DC has literally changed my life more than any other non-fiction book because the most important outcomes in our lives are the result of the most difficult conversations.
Your brother-in-law will eventually need to be forgiven for acting like a maniac, and you should anticipate that he will quickly transition into doubling down on the insanity because it seems less humiliating than admitting that he was acting like a tool.
Before the healing, however, you do have a problem to solve (or possibly, not). The first question is for you: is your brother-in-law making good decisions unilaterally? Is it possible that he's trying to protect his father from your wife? Don't be mad at me for asking. It could be possible that you should let him continue, as things are already stressful and perhaps having one person in charge is okay.
But let's assume that he's not making great decisions. What you need to figure out pronto is that when someone is upset, it's almost never what they say they are upset about. If your wife is pissed at you for missing dinner again, chances are that (unless you have an eating disorder) she's not worried that you're hungry. She feels disrespected, lonely and insecure about her marriage. So, if your brother-in-law says he's upset about X, the single most productive thing you can do is engage in creative empathy and figure out what he's most likely upset about. He's likely terrified of being abandoned by his dad, or perhaps is having a crisis of unresolved things that could never be said. He could be feeling powerless. The fact that he's the youngest might be manifesting as insecurity. (After all, why did you mention his age as relevant?) Perhaps he is terrified of death and seeing his father in this state is forcing him to confront his own death. I don't know what his deal is; just that it's almost certainly not what it seems to be.
As for the negotiation, here are some helpful suggestions straight from Getting to Yes: first, be prepared. It doesn't hurt to write down some of the following and maybe even practice it. Second, start every negotiation with a negotiation about the negotiation; "do you have a minute? I need to talk to you about what's happening". Third, be aware of what you want and what he wants, but also be mindful of what he is willing to accept as a compromise in combination with what you are willing to accept as a compromise. Fourth, you will both win better outcomes if you negotiation from your interests rather than a position. Why do you want what you think you want and is there another way we could possibly satisfy both of our interests?
It's critical to convey that someone does not have to lose in order for you both to win. (See https://www.youtube.com/watch?v=vdA2wecb4k0 for a further summary.)
A power move is to bring in a trusted but removed third-party expert to help with a difficult negotiation. It's important that this person is impartial and not arguing for either side or else the brother-in-law will rightfully feel attacked.
Finally, don't be so hard on yourself. I am proud of you for attempting to be productive using whatever techniques you have in a time of stress. Best of luck.