I was wondering if there is any way to increase the slots for your pet's treasure trove.

Yep! You can use they give you 1-3 spots and you can find them here.

are from the Cash Shop and cost 500 credits. I can't remember how many spots they give you, though. Will edit with that information in a second.

[edit]

Subeta Lodge says the Multi-Compartment gives you at leas 10 spaces, so yeah.

you can increase treasure slots with magical keepsake boxes

which you can buy from the millionaire center for 150k or off the shop search for 200k as you can only purchase 1 item there every so many hours c:

Thank you very much for your help guys. I will purchase those soon.

Also random events can increase them as well, also if you want to take a risk I know one of the zapper's can as well but it can also change your pet.

Zapper V2; the V2 Zapper located on Atebus will sometimes grant treasure spots.

Also the Loyalty Box (which you get after 2 years onsite, you cannot buy it) has, as one of 5 options: Increases all pets treasure chest by 10.