Mutt as in from Hunger Games or like a dog?

The Latter

Can you explain more about what you meant by this post?

If Teddy Lupin ever met Sirius what would Teddy refer him as seeing how there’s two relations that Teddy is tie to Sirius

Being Andromeda’s grandson Teddy is also Sirius’s first cousin twice removed but He’s also Harry’s godson

He'd have to refer to him as cousin since Andromeda is his grandma and she's Sirius's cousin that makes him Sirius's first cousin twice removed.

Or Grandma’s cousin or Great Uncle since Teddy would probably call Harry Uncle from what I recall in early Fanfictions plus Sirius would like an better title than Cousin from Harry’s godson

Great uncle since he's his grandma's cousin also this way Sirius can have another title other than godfather.

^ Also Sirius would probably demand Harry to call him Uncle since he is considered James’s brother