Actually I've never had a problem logging out with a pet and a full stables, but if you're worried you can always log in another char and check if the first one's pet(s) vanished.
Don't leave anything inside a pack animal though, that's a big no-no
Stabling the old fashioned way is still the safest option if you can manage it, but if you do use the autostable though, I'd suggest being in a secure room of your house so any wild pets can't run away.
Wenchy