Okay, I haven't tried this myself but it might work. You can do this if you don't mind cheating a little.
First, adopt a pet of the age and gender you want. Open the cheat console by pressing ctrl + shift + c. Enter "testingcheatsenabled true" without quotations, and then shift-click the pet. Select "Edit Sim in CAS" or something to that effect, and then you can customize their breed, traits, ect. from there. As long as you don't change the age or gender, you should be fine.