Badge/leader
>Insect Badge - Burgh
Team
>Trash Wormadam with infestation, metal sound, iron defense and flash cannon with high HP, average def and high sp def
>Vivillon with sleep powder, quiver dance, substitute and hurricane (standard-issue smogon)
>Armaldo with stone edge for muh fire/flying type coverage, sand attack for hilarity because why not, aqua jet and x-scissor
>minmaxed (high speed and attack) Pinsir with close combat, thrash, x-scissor and bulldoze
The gym puzzle would involve various infestations on each floor as you progress towards the leader.
One floor would contain a large swarm of combees blocking the way, with you having to pull a pot of honey around the gym to lure them away from different passages. Luring them away from one pathway blocks a later one, etc.
The next floor is home to a colony of durant, which block the various corridors with their marching paths. The walls have giant-light up heatmor likenesses, and when their respective switches are toggled, they light up, diverting the durant from their path.
The third floor is a lot simpler, two straight paths. One is flanked by pairs of battling trainers, so as to start a double battle when you get in the way and walk between them. The other leads you to Burgh, but is blocked by three pairs of battling pinsir, their horns locked together. After you defeat each pair of trainers, you gain access to a new switch, which activates a fan to blow a calming scent over to its respective pinsir battle, calming them and unblocking the path to Burgh.