LOOK at those sweeties...
When I first brought Chanel home they had a few tiffs - no actually biting but a few screams.... and it sure sounds awful...I did pay a ton of attention to Chanel so she wouldn't feel replaced or threatened - like, if I kissed the puppy Cheri, l'd kiss Chanel 3 times more....it really works.
I would always hold them together on either side of my lap and let them spend time there together too - eventually each learned to cope with the other being in close proximity... and who gets the best seat - (my lap).
We still have little Toy Tiffs..... even now, if they have a toy fight - they KISS each other almost before it even starts....So I think there's hope for you.... but if blood was drawn you do need to take measures that this doesn't happen again and expecially if you're not there.