A good to way to visualise how Tyson handles rivalry is the dual-meanings of the slang, “Come on, snake. Let’s rattle!”
He is competitive with Sebastian in a rival like way, during the times when he isn’t cowering from The Colonel. He pushes Sebastian to be the best possible version of himself; Tyson will drag Moran, kicking and screaming, if he has to.
The difference between working together and against each other is the difference between a single step in the dance, and back again.
Rivalry to Tyson takes the form of a rival from the Pokémon series. Not an adversary or antagonist, but a challenger; someone competing with another to push them to try harder and be better - competing with, rather than against - towards a common goal.
This is most obvious with his Love/Hate relationship with Sebastian Moran, but also applies to his relationship with the Moriarty Mirrors. It’s give and take. Maybe that means dancing over rooftops with Sebastian to bamboozle the police, or matching him blow for blow in an alleyway brawl.
Maybe that means joining the Moriarty Mirrors in whatever convoluted tango they have gotten caught up in, or maybe it means fighting tooth and claw to make them understand - and accept - a different solution when their gaze is already set on a shiny prize.
He doesn’t want to cage Sebastian or deny the Moriarty Mirrors their prize, but Tyson will never allow any of them to rest on their laurels or stray too far from the path. When he agrees with the goal, Tyson will match them step for step. When he disagrees, Tyson won’t dig his heels in and refuse to help. He will fight for a better solution, or for a justification for their decision, that satisfies everyone.
Tyson caught The Colonel’s interest and keeps the Moriarty Mirrors’ interests as their loyal Pup - not by being a ‘yes man’, but - by challenging them. He doesn’t disagree for the sake of disagreeing, of being disobedient and stomping his feet to maintain independence. Tyson fights to make sure everyone stays sharp and becomes the best version of themselves.
He doesn’t wait for an opening, a ‘does anyone disagree?’ or ‘are there any objections?’ Tyson will speak up whenever he needs to. He will bring the challenge to table himself: Come on, snake. Let’s rattle!