Paramnesia Buttwing Bomber and Paramnesia Recycle Beast seem to be difficult to battle even though Kem has passed into tier 13. I have googled them both and found nothing on either of these challengers. Are they something I should aim for or can I beat them now with specialised weapons?

both can be chain frozen so with the right amount of AP and sufficient scrolls, they can be taken down with any weapons. I think they do a lot of earth and physical, as well as magma, but I don't quite recall buttwings' icons, so that could just be for the recycle beast. unless they have loot past 10 wins, they both are still lootless and not worth the effort imo :x

They don't drop any loot. As kbbob says they're vulnerable to being chain frozen and that is essentially your best bet for dealing with them, particularly Para. Recycle Beast, but honestly its not worth much other than a sense of pride for conquering them. I was able to pull off five wins against the Para. Buttwing right after entering T13 earlier this year but it took me capping along with a hefty dose of luck, blocking, healing, and freezing to 10/10 Recycle Beast pre-reset. I recommend Ornamented Hook Swords(I believe it was) for the earth block, and investing in critical strike tear crystals to mod your weapons with. Also, pylonics and potions. Use every scrap of extra damage and you can.

is there any other reason than pride to get your battle pet a winner? Okay chain freeze it is.

Not really. You don't get any medals or achievements or prizes, just exp and a laughable couple thousand sP.

I was posing a rhetorical question and should have put a full stop instead of a question mark and the end of that sentence. Thanks for answering though. It did make me laugh. :D