I used to do a lot of regular gigs as a musician for about 4 years.
What usually justifies getting paid for live performances as a musician:
There is a paying audience that you are entertaining, and they are getting value out of the performance. In some way, on some level, the audience is there to hear you, whether actively on a stage or passively in the background.
What you are describing sounds like:
The venue is promoting an event and providing an audience and the performers are getting value out of the experience of performing to a crowd.
This is an open mic night, let's be realistic. If you are good enough to get a paid gig, you can organise that very easily. You did nothing to organise the open mic night, the venue did that for you for a bit of fun. It's essentially glorified karaoke. Would you expect to get paid to sing karaoke to your friends? If you ran a karaoke bar with a stage, would you expect to pay your patrons? It sounds to me like the entry fee is barely enough to cover the prize and I don't see anything that could be an unethical profit motive here.
The problem with this question is that it's dressing up open mic as a "gig", whereas I'd argue it is not a "gig" at all. In order to have a "gig" you (or your representative) either organises it yourself by contacting the venue, or someone representing the venue contacts you explicitly, ahead of time and requests that you perform on their behalf on a set date, for a negotiable fee. The other questions referencing formal documents from Unions and working musicians are missing that key point.
So, because open mic is not really a "gig", it is simply a fun community event with a fun community funded prize and while performing you have no responsibilities and there is no expectation for you to perform well or otherwise do "a good job (beyond your own ego)" in any way, I would say jump off your high horse and pay the $2 and have fun like everybody else is.
Edit: Even if you are awesome, touring and selling out huge venues nightly, it doesn't change the fact that the premise of open night is intentionally newbie/amateur friendly. If you are selling millions of tracks on iTunes, you could probably talk to the owner of the venue and get them to pay you $2 to perform, I recommend you give them a call to find out.