I'm going to be a lone voice I think, but you have asked for any opinions and perhaps it will be helpful on the off chance that this woman is like me. I know on here I'm not the only one 
You come across - to me - as patronising. So watch that with her.
You want to pay for someone else: you're an old fashioned gent.
She wants to pay for herself: you start deciding that she doesn't like to feel indebted, that she "seems determined to be self reliant".
Go go little lady for being - oh, a normal woman?
Maybe she pays for herself (where are you even getting these assumptions from?) because - that's just what women do.
Look, I may get called out by others - and you! - for being too harsh. But I genuinely mean it helpfully. I've done a bit of online dating and the "I pay because I'm a gentleman" type really puts me off.
I never want a man to pay early on - simply because, why should he? I actually find it unfair on men, I believe in equality and I want men to break what is to me, old fashioned crap, as much as I want to break it myself.
I'd find it really offputting if I found out a guy thought about - and posted about! - this stuff before even asking me out!
Just... ask her for coffee.
Offer to pay not because you are "old fashioned" or earn more. (if you even do) Offer to pay because it's a kind and generous small gesture.
But let's just throw this in; maybe - SHE will offer to pay. In which case, how about you just saying "thank you"? I had a couple of online date meets where I really didn't see a match because of equal paying / splitting. It's not "old fashioned" when you get to the insisting rather than offering stage - it's just rude. If she wants to pay, she just does. She's not "determined to be self reliant" - she's just an ordinary woman going about her day.
Leave any discussion about paying for babysitters til down the line a bit.
At 2 and 4, she may only feel happy leaving them with her parents - and have no cost.
By the way, I'm a single parent who happily enjoys her boyfriend paying for things sometimes, even though I earn 3x what he does. I love that he wants to arrange things and treat me sometimes. I wouldn't date him if he thought men had to /ought to pay. He'd just not be a personality for for me.