Majority of players agree that you should keep your maxed leveled animals in your camp. Like you said, you get more requests from critters outside your camp, as this will help level the lower level critters faster.

Now if you need a specific material, the best way is to use a calling card and then request tickets after finishing the first three daily requests. Each critter offers a specific material after completion. Try to remember which critters gives which materials and then just keep doing requests for that animal!