It is actually quite reasonable scenario, but it always depends on what you want.
Do you want to:
1. Be provider?
2. Be long term lover?
3. Be short term lover?
4. Just nail her and leave (alpha, asshole,...)?
5. Friend? Friend with benefits?
and so on...
Once you decide what you want you should chose strategy, set a frame and keep it that way.
The situation you described, she would start chasing him, but she shouldn't be chasing him because of money. She should be chasing him because he is attractive (fundamentals), exciting.
1. You invest into her emotionally (can't over do it), pay for meal(s), take her for dates and so on
2. Show strong sexual desire, start touching her, keep inviting her to your place
3. She MAY NOT be ready within 1-3 dates so she will say no. Here is where GC suggests to dump her. I held the same opinion but recently changed my mind. I wouldn't dump her. The reason is, if she is cute, hot and there are great vibes - what does it really matter if you have sex with her on date 1 or date 10? You don't have to be classical provider at all, you are just enjoying the vibes with her while pushing for sex, and waiting till she gives in.
4. Now you pull away, similarly as you described. I don't mean to completely ignore her, just become more passive, stop pushing for sex, stop inviting her to your place. But you need to remain positive and open, you can't become negative. She is still that great person like before - except now you calmed down and stopped investing into her. It is perfectly ok with you if she doesn't want more, if she doesn't want sex
5. Ideally she will start chasing you. You had a huge emotional influence on her and now you pulled back, she feels that she is losing the excitement, and will want it back. Thus she will come back after you.
6. Just keep your frame. You don't have to be short term lover, you can be long term lover, and it is ok if she knows that you won't be a provider
I went through similar scenario just about 2 weeks ago. First date, nothing, just mild vibes. Second date, I started to touch and push, I was relentless, pushing strongly for sex, inviting her to my place. She refused several times but I was totally fine with it, just remained open, vibing, friendly and positive - then I just dropped the whole sex idea. Since then, she is contacting me every single day, telling me that she made a mistake, telling me how great would it be to be together at her place... When I see her next time, I will have a free ride. No effort, just show up... I paid for her dinner, and there is no way she can see me as a provider...