How to interact with an irrational coworker - #1: Fake Deals

A friend of mine asked me for advice on how to deal with a situation he's involved in as an employee interacting with his coworker who is acting irrationally.

This is a scenario where there are 2 job roles, where one of them is the money-maker and the other one supports him/her. My friend is the money-maker and his coworker is filling his support role. Each money-maker is assigned one support person.

There was a series of previous cases of irrationality which led to my friend discussing it with his manager and making a request to reassign his support person. (IIRC there was no clear answer from management about the request.)

In the latest case, the coworker acted as if a deal was made between them when there was no such deal. The interaction happened in a chat group which included the managers. The coworker said something like, "hey we had a deal!" in response to my friend not agreeing to something. My friend said, "what deal?" And then the coworker quoted herself from a day earlier. 

I initially gave a recommendation to say something like this:

i think we may have a misunderstanding. can you show me where i said that i agree with your proposal deal?

My friend didn't like it. he said that it was/is clear that no deal was made. so i gave the following recommendation instead:

a deal is a mutually-agreed upon thing. that means at least 2 people involved. why did you only quote you talking?

I clarified that talking like this might get him fired if the company is bad -- which I argued is a good outcome (which factors in what I know about my friend's particular context) -- or promoted if the company is good.

The coworker didn't address the question and instead said that they want a meeting to discuss these issues. 

My friend thought he might get fired and wanted to avoid a meeting if the firing decision had already been made. So I wrote another reply saying something like: 

Dear Staff/Management, 

If you've already decided to fire me, please do it now to save us time, else I'm happy to have an honest and rational discussion about how to move forward.

The coworker replied with something like this:

I only care about the issues I raised, I don't know about all that other stuff you said.

So I wrote another reply for my friend like this: 

I understand that you have issues you want addressed during the meeting. I also have issues I want addressed. So management must be present and involved in the meeting or else I don't want the meeting. 

The coworker replied in a very interesting way. She completely changed her tone. She went from bossy to polite.

And immediately after her reply a manager chimed in saying he will be at the meeting.