What would you have me do?

Is there a name in rhetoric or fallacious argument for this type of thing? 

What would you have me do?/What else was I supposed to do?  etc.

It seems to be an attempt to shift responsibility, probably also a distraction from the main argument.

This was used after a vote was put up on a group chat. The vote was clearly a false dilemma between two options, neither of which were things I wanted, although one was presented as my  preferred choice. I'd clearly stated that I wanted a discussion and not a vote. There was no reason to vote between two options as many alternatives were possible. It was an attempt to avoid open discussion. The vote was also rigged in favour of one option. I left the group and received a message with the  'What would you have me do?' line. This all followed a previous message which I've discovered used a truly ridiculous amount of rhetoric and fallacy so I'm presuming this is also some kind of trick.

Thanks.

It is like implying that you had *obviously* absurd demands and you were *obviously* unreasonable. Like telling you "I've done all I could. WHAT MORE DO YOU NEED FROM ME???" That could be an honest question, but if you already had explicitly told him what you want, then I think it's something like:

Appeal to Exasperation = your demands are *obviously* so unreasonable that makes me lose my mind (usually said in an emotionally loaded voice or shouting in exasperation), implying that 'you should be ashamed by your unreasonable demands' and basically accusing you of being unfair for accusing someone who in fact did everything they could. 

That's the closest I can think for what people, in many cases, use to silence one's complaints - being at fault (or insincere or mistreating another) and then acting like they're the victim. Like Baiting and Bashing  which is a similar technique used by abusers, narcissists etc.

It can be also what is called Emotional Reasoning , when they don't deflect but truly believe that since they have an emotional response (being angry, sad, mistreated), their feelings accurately present reality and their feelings are proof that the other is at fault.   

Although this is not a form of argument but a question, it exudes a spirit of unverified claim that the other side has done everything in its power and that any request beyond that is superfluous, if not rude.

Of course, this question in itself does not have to be malicious, someone can honestly ask this with the intention of hearing the other party's suggestions about a possible change of actions, but this question is often misused for the purpose you described.

There is no argument here; it is just a question, so no fallacy. It might even fall under Socratic questioning if the question is sincere.

Sounds like a question only Shakespeare can best answer.