Sorted by rarity - i check only once a year for any rarity changes, and move a handful of pets. I make the page size to the number of pets in that group when checking rarity, and it takes maybe 10mins max. If I have any rerealease pets I put the original month/year of release in the name.
Looking at others pets for trade, I only use the "pets on your wishlist" option, but if I notice they have loads of groups with NFT in the name I don't bother trading with them at all. If you don't want trades on those pets lock your groups people!