Personally, I'd rather Fai actually WAKE UP before anybody bandages him up. Even if that's maybe a little mean of me. Though really, he's bleeding but it's not as TERRIBLE as it could be. A bandage might not do much at this point.
I'm thinking... maybe CLAMP has kept Fai unconscious for this part for a reason. Maybe they don't want him reacting to the bat symbol on clone!Syaoran's shirt too.
I've always had my suspicions that Fei Wong Reed has had something to do with Fai's past since the little-Kurogane story arc.
Perhaps Fai even had something to do with Kuro-mumma's death by chain of events. Maybe he's aware that something he's done has effected Kurogane and that's why he's so desperate not to divulge any information about his past to him.
My theory is that Fei Wong Reed had contact with Ashura. Ashura somehow had Fai under his control and used him for his magic. I'm not sure... but since Fai has the power to create life, maybe he created the Oni that killed Kurogane's mother?
Fai could have tattooed himself with that magical device to suppress his powers and stop them from being used in such an awful way. That would explain why he's so reluctant to use his powers anymore, as well as why it was his most precious possession and why it was also the seal on Ashura's 'coffin'.
It does kind of make sense that Fei Wong would have a part to play in Fai's backstory since he has in Kurogane and Syaoran's.