Generally the best way to do this is to use a ViewportFrame. Each button has a Viewport frame with the appropriate pet model in it, and it will give you exactly what you want. You can even make them spin as a bonus. A guide on how to use them can be found here:
Edit: Although for some reason it tells you to make it entirely out of code - you can insert and edit viewport guis like any other gui, you just need to be able to set up the camera. Here’s a more to the point guide: ViewportFrame Release