Actually the question does still hold interest IMO because it forces her to acknowledge to herself that she may have a price.

This is ridiculous. The question does make her uncomfortable, but not because of the question itself. She knows she'll sleep with a guy for a million dollars. She's thought about it. She may have even thought about her "price" for letting a man fuck her. The problem is she knows the limits on what she's supposed to say and how she's supposed to present herself, and here's this clueless jerk trying to force her beyond them. She's playing the game, navigating social rules, and he isn't even acknowledging the tight spot he's put her in. In fact, he's simultaneously relishing her discomfort and resenting her for feeling that way. What does she do? To get through the situation as easily and harmlessly as possible, she either denies the legitimacy of the argument or names absurdly large amounts of money.

To really get a woman's price, she'd have to be assured of discretion. And her price would, ultimately, be affected by her confidence in the assurances of discretion offered. Her price for prostituting herself openly would be much higher, though not as high as the price she's willing to admit to in the original joke or the experiment you describe. After all, in the experiment, she pays the price of advertising her willingness to have sex for money without actually getting the money. A million pounds buys a lot of honesty -- what were you offering? Even with the check, you haven't established a credible offer. The woman would have a lot of doubts that would be difficult to overcome. Is it a scam? Why me? Does this guy want to hurt me or humiliate me? Is this a mean-spirited prank organized by one of my exes? From her point of view, it's vanishingly unlikely to be a genuine offer. She's thinking, "If I even give this guy a chance to prove his bona fides, I'll probably be putting myself at risk." After all, even if he actually has a million pounds in his checking account, he's still more likely to be a killer than a guy who pays a million pounds for sex. Just sayin'. She's not stupid. And if the guy is serious, the onus is on him to prove that he understands her reservations and to think of some way to reassure her. Her initial response shouldn't deter him.

Anyway, to depart from your experiment and get back to the conversation in the joke as it's usually told -- no longer talking about your experiment -- it's a typical conversation for socially incompetent young geeks who are frustrated by all the social taboos, who suspect (partly correctly) that everyone around them is screwing like rabbits, and who are so painfully frustrated about not being able to talk about it that they make fools of themselves beating their heads against the wall of taboo with rational arguments instead doing something that might actually clear the way to frank conversation, such as cultivating trust and intimacy. (Gosh, I just might be speaking from experience here.)

So the whole thing resolves to a guy making a girl uncomfortable and taking her refusal to be outré for stupidity. That's pretty dickish. Especially when the point is to make yourself look smart in comparison. I have NO idea why Feynman liked this joke, except that maybe he used it as a racy line-crossing move when chatting up women, in which case the logic itself is kind of beside the point. I supposed he knew the right moment to push it. Or maybe he just knew his audience. But that would make him a bit of a misogynist, since he would know -- admit it -- that the chief delight of this story for most people is not whipping it out Feynman-style at just the right moment when a woman is ready to let down her barriers. The chief appeal for most is getting the last laugh on a woman who wouldn't let you past her facade of propriety. Why would he stoop to that kind of pandering?

Anyway, the original word game breaks down if you examine it just a little. If I'd bake a loaf of bread for a million dollars, am I a baker?

If I'd do your taxes for a million dollars, am I a tax accountant?

If I'd write a book for a million dollars, am I a writer?

If I'd change your oil for a million dollars, am I an auto mechanic?

If I'd teach a yoga class for a million dollars, am I a yoga instructor?

So is she a prostitute? Clearly not. Is she a whore? Well, yeah, quite often she is, in the sense that the word "whore" in any language usually means a woman whose sexual activity makes the speaker feel bad in some way.

But don't worry about this line of reasoning, or any other, being used against you when you whip this gem out at a party, because the response will divide between a couple of straight-up misogynists enthusiastically backing you up and a majority who just distance themselves from you, possibly by scoring points off you in some irrational way that is COMPLETELY IRRELEVANT to the point you're trying to make, because people are STUPID and more interested in playing STUPID SOCIAL GAMES than actually THINKING. So your point is unchallenged! Unrefuted! Whoohoo! (Sarcasm aimed at the original joke and at my teenage self, not personally at you.)