I really believe it is the rough play triggering it, and I bet she was giving signs (growling isn't always the sign, whites of eyes, position of tail, etc can be) but I fully understand. If your children are older than a toddler, you want them to be able to interact and not be separated all day until you can specifically monitor. Not every dog owner (including children) can spot such subtle signals before bites, it isn't worth having her with the chance of another bite.
Personally, I see this bite, you keep the dog, behavior continues, another bite, children treating the dog bad because it bites, etc. It will just turn out bad no matter what.
Try and find a good home for her and don't drop her off at a pound/rescue. Your situation isn't fatal, she can be separated or monitored for a bit until a home is found.
Fact is, some dogs have higher tolerances and better bite inhibition than others. This one seems to have mild tolerance, and by what you've said, you need a high tolerance dog who isn't going to snap even when being harassed. You'll find it!
Kaida- CT Female 2.5 Gal.
Julep- HMPK Male 2.5 Gal.
Fandango-HM Male 10 Gal.