Here is a quote from Garg's link that applies directly to this issue, Claire:
"Federal copyright law does not apply to cover bands, said ASCAP spokeswoman Cathy Halgas Nevins, only to businesses profiting from the music."
So even according to ASCAP, you have no liability here--look at the dollar amounts mentioned in the article:
"ASCAP is seeking $750 to $30,000 per infringement, the complaint said."
If that band had signed the agreement that your dude is trying to get you to sign, they would have been responsible for that amount of money(or for proving, possibly in court, that they didn't infringe on copyright--which is where legal fees come in).
Yes, you will get to play, but there is a chance that you'll have to pay, as well.