Most young animals have a crazy hour or half-hour just before settling down or the night - nature's way of making sure they are warm enough to make it through a night outdoors.
So this would be a good time to take him outside.
However, not even a young puppy should be allowed to bite or snap, so reprove him gently when he does so. The sooner he gets the message that biting and snapping is not acceptable behaviour towards humans the better.
"Bad boy!" said firmly may be enough, but you may have to resort to a light tap on his nose (not to hurt, ,just to make the point, although the cats, as you are aware , will slap him hard enough to hurt if he snaps or bites them)
As he knows cats hit out, and have claws, hissing at him as much like the dominant cat as you can manage, may very well be the best way of ensuring good behaviour.
A good friend's three month old labrador is unwilling to comply with commands such as "Sit!" when told to do so by us, but when my Tigger arches his back, fluffs out his tail to the size of a flue brush and walks sideways towards Molly, she sits down on her behind very promtly. And as yet neither Tigger, nor his sister Trubel has slapped or hissed at the dog.
Trubel just employs the look common to the elder generation of school-mistresses in my childhood.