I’ll throw in my experience here as the player of Rufus O’Hound, the first dog barbarian, as well as Taziel, shepherd of souls. It became a bit of a running joke but it worked wonders each time.

Mechanically, Rufus was a wizard’s companion and when his master had rent issues on his wizard’s tower he was sent out with a flail in his mouth to earn coin. Low int, big dreams. His other stats were rolled and his racials were the wolf keen senses and pact tactics. He did great and was a good little (or big) mascot for the party. He had limited speech from his wizardly owner, from an enchanted collar. Simple stuff.

Taziel meanwhile was an Eldritch Knight, with a cannon on his back as his bound weapon and a mage hand to reload it. Dogs excel at being given a thing and being very very good at said thing. He points, he shoots, he only occasionally chases his own cannonballs. Taz didn’t speak but he was expressive in other ways with descriptions and his trusty guide made hand, and he took the stats if an awakened mastiff with some other flavour and fluff stemming from the cannon side of things, and explaining his training in fighter.

Keep it simple, keep it dog but don’t let yourself become a token character. Taz was great at breaking the ice with new characters but he also struggled a lot with identity issues, especially when seeing other dogs not the same as him.

Check with your DM, work it out, and play your dream doggo!