Discovered Blood during a TO

[Scooby]

This happened the other night. A1 was injured during a play. She showed signs of a concussion. As she was being helped back to her bench I saw blood on B1's calf and on her shorts. I asked my partner to inform B's coach. B's coach said that he should be given time to fix the blood because we should have caught it sooner. (Nice try). He was very insistent about keeping her in the game.
So the question is if she could have cleaned up within the injury time out could she have stayed in the game without B's coach taking a time out?

Rule 3-4-7 says the player must leave the game unless the the players coach is granted a TO.

Are there exceptions? If you see blood on a player heading off the court for half time would she have to sit the beginning the the 3rd quarter (seems harsh)?