Well, it sounds like he is doing some things right.
- He pays if the kids aren't along, and for things outside of dinner/meals, just not those, am I right in saying that?
- He enjoys your family and accepts your kids, even interacts with and mentally supports them, just not financially or emotionally. That may be a matter of principle for him.
- He makes a point of spending time with you, quality time, and consistently it sounds like?
So I guess here's what I'd say. It is a bit weird that two months in he's pushing bills off on you and not at least offering. It may be a test, which is dumb. Maybe let it go 2-3 more times, but after that push the bill his way next time and say, "your turn!" If he questions it that's kind of bullshit because he clearly has no issue doing this to you.
How much do you like this guy, and are you seeing other people? If you aren't exclusive, have the time, and actually have the interest I'd start seeing one other person maybe - just to see if you like it any better or nah. I can't tell how serious an issue this is for you, or why it bothers you, and maybe you don't know. If you're just starting to date again then take some time to figure out what you like, who is out there, who you connect with.
How would you say the rest of the relationship is?