I would say whatever you're comfortable with if it's not in your divorce decree. If it's in your divorce decree, I would follow the rules. If he can still go out to dinner and do other things when he's not working, and not pay child support, then I WOULD NOT allow him to claim her because he doesn't deserve to. I also wouldn't allow him to claim her if you're providing anything above 70% of her care, including a home--her own separate room, utilities, food, clothes, lunch money, internet for her to do homework--very necessary, gas to take her to school, doctor appointments/medicine, etc. Some mothers might think it's petty to list those things, but none of that is a luxury; being a single parent on one income isn't easy and every $ comes in handy. I'm sure if you didn't need the child support, you'd probably not worry about whether he paid it or not. Even if he keeps her 2 weeks out of the year, that doesn't warrant him carrying her as a dependent because you provide a home with a room for her, whereas when she's with him for short periods of time, she probably sleeps on the sofa. I definitely wouldn't allow him to carry her if he's behind on child support and didn't bother to help in any kind of way while he wasn't working. That doesn't even need to be pondered. As far as the extracurricular activities, I wouldn't worry about him paying that. Getting the child support is all I would focus on. When she's old enough to understand and make decisions for herself, she'll appreciate all that you've done for her.
I can tell you from personal experience that if it's not in the divorce decree; she lives in your home the first 6 months of the year, and he carries her in addition to you carrying her, he will be in trouble with the IRS. My ex had a verbal agreement with his ex that he would carry one of their sons and she would carry one. Unknown to him, she carried them both and when the IRS caught up with them, WE had to pay $10k in taxes. For 7 years we never received a tax return. Before we were married, I received a return every year. Think long and hard before you make the decision. Don't allow him to carry her if you'll end up regretting it in the end, because once the forms have been filed there's no turning back. If it'll make you feel better, you can always share some of the money with him that you get back from the IRS.