I think the hawk in question would object to the idea. Worth a shot to ask once, but I'd say it's a no-go. Besides, what's better? A loony hawk who is taken care of by an intellectually superior wizard and a band of adventurers, or a loony hawk living completely by himself in the house of a spellcaster? Or should he ask his family to take care of this avian of the century? That would be hell of a visit. 'Hey, Mom, long time no see, sorry I didn't Message. I was wondering, could you take care of Lola while I go kill monsters in someone's oversized basement?'Leave your hawk at home.
If you're just wanting to get rid of the hawk so you don't have to worry about the potential loss of EXP, then you won't be getting a new familiar anyway. No reason not to just leave him at home.
@Neonchameleon Curious predators often put themselves in situations where they don't see a potential danger, rather than choosing to ignore it. Birds especially.
@Dozen The one where the hawk isn't in imminent danger throughout every day, where spells, arrows, axes, and giant clubs aren't the norm.
One question left. What's an acf?

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.