Its a rough situation vise and I can empathize with you. While its not likely to help your case to fight over a few dollars here and there the bigger issue is the way she keeps attacking you knowing you will back down. You are going to have to stand up to this type of manipulation and disrespect and get a footing to become a new, strong man but without being a cold [censored] about it. Sounds impossible, and it will be difficult but you can do it.
You'r trying to stand up to her but its not working the way you want it too at all. I'm not sure going through each and every receipt is the best way to get where you want but you do have to find a way to stand up to her without backing off the second she shows some claws and not melting the second she starts getting emotional.
I feel for you, this isn't easy.
So, she used this excuse of looking out for you since her parents have extra money and can help her. This is going to be held over your head since in her mind shes the one doing whats right (helping you)and your being the penny pincher who's trying to control everything. You can politefully refuse that help and explain you didn't ask for anything to be handed to you or for her to be nice, that all your looking for is to pay your fair share of everything and having an understanding of where the money is being used.
She really does need to have her account separate from the kids account. If you need to maybe you should offer to open a new account for the kids money that you both have access to if her moving to her own account is causing too many issues.
The situation obviously isn't working and causing conflict, so it needs to be changed. How you do that might cause other conflicts but its something you need to do.
If you need to pay more so that theres less conflict and things cant be used against you, then so be it. If you cant pay more and this is what needs to be done then you still have to find a way to stand up for yourself without being a dick.
Accept what is, let go of what was, and have faith in what will be