I'm guessing it's a bluff anyway. Presumably they have their own money on the line, so sabotaging their investment is against their own interests. If the CEO caves the angel walks away with more equity for nothing. If he doesn't, no harm in asking, right?

I think the appropriate response is to thank the angel and be appreciative of their contributions but explain that there is no more equity to give.

I'll rather close than keep working with someone that's playing that game. Even if they pull back that maneuver, I'll have to think a lot to sign that round.