Leave him - he is not a keeper.
To all those instantly jumping in to say ooooh the dog would come first... the way I see it, the main reason I'd walk away from a joker like this is precisely because he doesn't put his dog 'first' in any meaningful way... or even care for it adequately. I absolutely HATE so-called dog owners like this - lazy, selfish, see their dog as literally something to be entertained by and fawn over but they're not willing to put in one single second of hard work to make that dog secure, socialised, happy or safe.
Poor dog. Under-socialised, frustrated, barky, aggressive, probably doesn't know whether it's coming or going with either other dogs or people as it's never been trained properly. With an owner who thinks loving a dog is fussing and sloppy kisses and getting all twattily 'protective' and defensive even when that dog - because he's made sure the poor thing knows no better - actually bites someone.
The kind of shit for brains 'owner' who would be there in floods of tears at the vet's the day his dog has to be put down because it's scarred a child for life. Yep, dogs first every time, wahey! What a guy, putting his dog first! How dare you, OP, come in and be nasty to poor doggy woggy by trying to set boundaries and treat the dog responsibly...
I would get rid because the way this arse 'cares for' his poor dog tells you a lot about the kind of man he is. Full fuss on looking as if he's so caring and lovely but won't for a moment want the hard parts of loving - taking responsibility, making hard decisions. Wants to be the fun one, and turns a blind eye to when his 'fun' actually means unhappiness and inconvenience for someone else. Irresponsible, entitled, unreliable.
He'd be the same with your children. You've heard of a Disney dad - this is the Disney dog owner version!
So yes, dump him. Unfortunately his poor dog is stuck with him, but you aren't.
Oh and for a non-lover of dogs, what you've said so far shows you to be far more suitable as an owner of one than him!