I have to say mooch as well. It is not about him having money, but he seems to have expensive tastes and be sneaky by only saying he has no money when the bill comes. I would think somebody who was not a user would be trying to find creative, fun ways to spend time together (or at least be upfront about the money situation and let the bill payer choose the restaurant). If this guy chose a fancy restaurant for his SO's birthday and expects her to pay for the big night out, he is really showing no effort in making her birthday special for her. If I was in this woman's shoes, I would feel much more honored if the man would plan an inexpensive special date for my birthday, then choosing a fancy restaurant with the expectation of getting a free meal. I don't think a person can turn off resentment like a faucet and the OP will wind up resenting the situation if her date left her with the bill again. He may be willing to surprise her by springing for dinner, but it isn't worth taking the chance of a ruined evening if he didn't, especially since he has a reputation for not paying.
I believe that this can be an awkward conversation to have, but if a person would blow up or get upset that their SO did not want to subsidize their fancy dinner plans, then good riddance. It is better to discuss and know where she stands than to continue on and feel used and resentful.