Are there any advantages to having your pet in the resort?

For sure! This way, you don't have to worry about keeping your pets full and happy for features like the Battle Coliseum and games. If you don't use features like that though, then it might not be worth it.

Just a note that if you do Major Drills' quests, it's helpful to keep one pet without a minion and out of the resort so you can feed/play with them for quests.

Thanks very much!