In the event the kids get hurt/sick while they are with her, she needs to be able to get them treated. It is getting harder to just go in with the info. I watched my dr's office of 15 years REFUSE to treat a college kid who only had the info. Even though the beauty of BCBS is the ability to usually just give them id number, cause every thing is filed on the local bcbs then routed.
You should still get a copy of the eob's. When you do just make sure they are for a valid trip. If not call the insurance company and explain that bio mom has a copy of the card for the kids benefit and you do not beleive the charges are actually for your child.
Also, the insurance company can usually provide a card JUST for the kids and not everyone else. If not use a sharpie and black it out. You can also add info such as children DOB to the card you give mom to help aleviate deception.