This took me a while, anon. I had to chew on it. I mean, don’t get me wrong, I love worldbuilding questions so thank you, but this one was kind of difficult for me to wrap my head around.
The thing is, the Arm in Arm world is more analogous to the Naruto world than the Pokemon world, except Pokemon basically take the place of chakra. So the clans wouldn’t really have type gyms so much as they’d have clan Pokemon (Nara with the Sawsbuck, Uchiha with the Talonflame, etc. etc.) So it’s more like the clan Pokemon are the clan bloodlines than gym types.
That being said, I do like the idea of gyms somehow existing even though that would mean a lot of gyms all in the same village. So they’re probably not full out gyms, but they are…
Oh. The different departments, probably.
Sensory Squad and Intel Division and T&I and ANBU, etc. etc.
So it kind of lines up more with DoS canon, in that after the genin have been trained by their Elite Four-jounin, they join different departments for additional specialization and training.
Sensory Squad trainers definitely use their Pokemon differently than Intel Division trainers do, but given that shinobi-trainers are meant to handle all kinds of missions, you can’t just be like “I know how to battle, I’m a master now.” So you can temporarily or permanently join a department/gym and get more rigorous training for you Pokemon in exchange for doing a number of missions or something like that.
Arcanine would definitely benefit from having an Inuzuka teacher. Or when Staryu evolves into Starmie and suddenly gets Psychic abilities it makes sense for Shikako to seek additional training since it’s not a type she’s previously had, so she does a rotation with the Intel Division. Stuff like that.
Yeah, it’s not so much a “battle your way through and get a badge and qualify for the League tournaments” seeing as how Arm in Arm has the genin teams training with the Elite Four right off the bat.
Then again, maybe for genin who graduated but did not get put onto a jounin/Elite Four team that is what it’s like. You know, do a rotation in each department/gym and if you complete the set then you can apply to be put on a temporary genin team for the Chuunin Exams?
Hm… still not 100% on this though, anon. But that’s what I have for now…