Dear David,

In your question, the man hired the woman for Z'nus. Therefore, he has a monetary responsibility to her for her job. It is his obligation -- not because of his promise of words or because of a Mitzvah -- but because of his contract with her to pay her. (Of course, the Torah commands to do the right thing and not to steal, so you can say, loosely, that there is a Mitzvah to pay.) But here, at the same time that the monetary obligation starts, he is guilty of death for the Z'nus, so Beis Din cannot force payment, but he nonetheless is supposed to pay.

Mitzvah ha'Ba'ah b'Aveirah refers to carrying out a Mitzvah by doing an Aveirah, like stealing a Lulav to use it for the Mitzvah, etc. But here, after the responsibility of payment exists, he must do his responsibility and pay.

All the best,

Reuven Weiner