Breeding new pets?

I’ve recently spent some Fate on unlocking Flute Street, which has been fun. Without posting any fate-locked information, I wanted to know if there’s a way to breed the pet from Flute Street into something else, or is that only possible with the Fate-locked continuation of the Labyrinth of Tigers?

It’s possible to breed it the usual way without the fate-locked labyrinth bit, but that’ll just get you items like candles or gossip or whatever. If you want new pets, you need to pay the fate.

Without the fate-locked portion, the hound is basically identical to the somnolent hyena.