If you give an item-using pet an amulet of unchanging, they will recognize that you want them to wear it and put it on.

Shapechanging monsters are cognizant of when they’ve turned into a form that’s much stronger than their usual; if they happen to have an amulet of unchanging, they will wear it when in such a form.

Assuming an implementation that lets monsters wear rings and lets you pick and choose what your pet equips: You can have your pets wear an amulet of unchanging to make them immune from being polymorphed or shapeshifting; possibly the rings of sustain ability could protect from both as well, and having them wear protection from shape changers could be used to prevent them from shapeshifting, if they are a shapeshifter.