Personally, I set my pet loose and just get a new one after Pesach.
About the rabbis answer, if you sell your pet who will feed the pet its chometz food, now that the pet is owned by a goy.
In any event, what kind of rabbi is this where he has congregants that commonly have pets, so much so that he has to speak in shul about this question? Very few frumme Yidden have pets.