I think we should have a sort of stable that we permanently put pets in that we don't want in.
Like, there would be a button on the Sylesti's page, and we click it and send the pet to the stable. If the pet is in the stable, you can't breed it, battle with it, use it on missions, etc. It would be an alternative to releasing and never seeing the Sylesti again. You would be able to visit it whenever, and it would have unlimited stalls and you could move the pets around in the stable.

I don't really see much of a difference between this and releasing.

You can "visit" released pets anytime through advanced search, and you can even reclaim them at any time (though the price can be a little steep, especially for newer users).