@tjenatjabamoss I do this through Manage Households. I just move the pet over to the other household that way. The Sims that are left behind show no ill effects, in fact mine don't react at all.
I started doing it for my dog rescuing Sim as she adopted too many dogs for her household. So I started adding them to her friends and family's households instead.
I just go to Manage Worlds, select the household then Transfer Between Households. Then once the pet is moved over I go to their new home to arrange a welcome and set up pet items for them.