I would stow mine and re-summon it.
It doesn't take long at all, but keep in mind there is a daily cap on how much blood it gets.
At first it was bugged but Rhet fixed it, so now it actually works.
Once you reach the total needed blood for each tier, then I'd just pray to tier up.
Rinse and repeat with the cap, you don't even need to hunt with them to bond them