Hooker Math is always entertaining! Pitching out $2,000/2hours when her rate is $500/60 minutes is justbasic market haggling. She probably didn't even realize that the numbers didn't add up, she was just starting highwith a random number, assuming you would haggle her down. Don't disappoint the lady . . . haggle!
First off, you can pay at the end, the girls are used to it. If she asks for it upfront just smile your charming best smile and say 'at the end' and maybe hint at a tip if she's really, really good. The alternative is to go half and half, but if she's that uptight about it you might be better off walking away.