Dodger has been aged by his experiences and he'll try to play that card.

Dodger sees physical damage as a sign of survival, so Reaper's appearance is both endearing and inspiring to him. Plus... he considers himself to look pretty fucked up, so getting to be the pretty one between them is nice.

And... Dodger's kind of exactly like a dog, he gets uncomfortable and upset when he doesn't have a boss to keep him in line. Romance aside, he'll be very happy acting as Reaper's helpful shadow.