I honestly think you should. There are plenty of men out there; yes, even men willing to marry a woman with children. Caring, loving men who don't cheat. Talk to friends and family; I'm sure your children will be fine especially if they can get one-on-one care and a friend's house or a relative's.
If he's been in counseling and cheated WHILE in counseling then it sounds like it's not working... or he's using it as another way to hook up.
His being bisexual is a non-issue to me. So what? Nothing wrong with being bisexual; if two partners have decided to marry they should sleep with each other exclusively no matter what their genders are.