at what point does she offer to pay?
Wait, this happens? Don't think she'll offer anything along those lines. I've found that despite all the feminists whining about gender equality, the man is usually expected to foot the bill, in line with traditional gender roles of inequality.
You might be able to set the expectation of alternating responsibility for the bill, "I paid last time, so could you get this one?" If it works once, you may be able to give the trend some momentum. Results may vary by woman. Good luck with that!
While I'm posting, I'll share my philosophy on this. I hate the idea of a woman serial-dating for free lunches, so I typically don't pay for anything until a third date. This approach can address the adverse selection problem for women who fit the "independent woman profile". However, if a woman fits the "domestic woman profile" and I really like her, I'll pay earlier and consistently to signal "good provider".
I realize there's a personal budget constraint that explains your initial post, but what is your sense of your date's profile? Does she hold a job and have aspirations for going somewhere in her career?
If so, paying her way may not be as important and you might risk proposing she chips in.
If not, not paying her way is risky; it may signal "bad provider".