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>Type
Normal to start. In rematches, any Pokemon capable of learning TM by level up or TM would be in the team.
>Gym Puzzle
Nothing particularly interesting. Might do a mock-up of a Zelda dungeon puzzle in a vague reference to the Triforce
>Team movesets
Both Pokémon spam Tri Attack, the other moves don't matter.
>Badge
Badge would look pretty much identical to the triangle symbol on Togepi line and the Latis. Name really wouldn't matter.
>TM
Tri Attack
Later in the game I'd pop up and gift the Trainer a Tri Attack Togepi (callback to the Togepi in XD, because Serene Grace Tri Attack Togepi would be a fun son of a gun in-game)
Tri Attack is my favorite move. And some more vague references to the implication Porygon is a duck, the shared triangle symbol in design between the Toge and Lati lines, etc. would be fun. I also enjoy the idea of a Gym based around a move vs. a type, and a callback to Gale of Darkness is fun.