There are far more people on this site trying to collect a complete set of every pet, than there are complete sets of pets to be collected. Even if someone is just trying to collect all of a specific lineart, that's still wildly difficult, unless it's a newer lineart. It also means that there are a lot of pets sitting in various groups, not because the person specifically likes that pet, but because they need it for their collection.
So, how about a way to collect everything, without having to keep it?
When you get a pet and the archive entry for that pet gets the "you own this pet" notation, it should remember that even if the pet is traded away. For example, say I get the 2009 Malk frankenstein-style horse, then decide to trade it for the cat from that same event. The archive should then display a little notation that says "you have previously owned this pet" on that horse's entry, so I'll have a record of it. Kinda like how the Pokedex, in every Pokemon game I can think of, shows whether you've caught a certain 'mon regardless of if you currently own it. Heck, maybe add a new archive subsection aside from "archives" and "your wishlist" called something like "your petdex", so you can easily browse whatever you've had.
This way, people can build a full collection, at least of a sort, while still being able to trade the pets involved. It would free up a bunch of different pets for trading, and would give people a more attainable option for collecting everything. Plus, anyone who wants to build a full collection of pets they still own can still do that, and they might be able to do it more easily with people likely more willing to trade rare pets.