If you are recording the band's live performance in a bar or auditorium or some such venue, the band has no obligation to pay anything to perform the songs. The "performance license" in that case is paid by the venue itself via a yearly ASCAP (in the US) dues license. and there's no obligation on the part of the band whatsoever.
For the recording, however, that comes under what is called the "mechanical license", and yes, the band is responsible to pay the royalties to cover that, and yes, if ASCAP wished, they could and probably would pull you into it as well.
If you are just making a recording for the band to distribute for free demos on a limited run, while still technically responsible, ASCAP probably won't care much about that at all. If/when the disc is retailed or resold, however is when they'll sit up and take notice.
While they probably won't come chasing after anybody of their own volition unless the recording has fantastic sales figures, all it takes is one disgruntled drunk audience member to decide thy are going to get even with the band for kicking him offstage, to report the situation to ASCAP in retaliation. While unlikely, it has happened more than once. This would probably generate a letter form their lawyer requesting either a cease and desist request, or a request to pony up for sales so far - along with a sales report from the accountant - or both, with a warning that if they continue down the current path and don't straighten up after that, they and you could then be threatened with a lawsuit. Agreening to stp sales and pay a small royalty sum or settlement fine would resolve that situation with no more ado, probably.
It costs them virtually zero time or money to send out that letter, since their lawyers are on retainer anyway; the lawsuit would only come if the band flipped them the finger and just kept going without stopping or reporting sales, and only if the sales were enough to warrent their time and energy.
Chances are slim that they'd have trouble, but it's like car insurance. The royalty fees are so minimal, that it's stupid for the band not to just do it right and pay them if they want to sell the CDs. If they don't, I might consider handling that yourself, and folding those fees and an extra fee for your own labor into the recording charge, just to cover your ass. That might lose you the recording gig, but it'd keep you safe. It's totally up to you as to whether you want to take the odds. Best bet it to just tell the band to do it right.
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G.