I don't know what we could call this particular breed, but alongside Artefact and Spirit, it would create something different in the game - and of course make them immune to stamina drain/leak/devour.
Monsters that have 0 stamina costs on their moves are not necessarily overpowered, as they can still be just as easily killed as regular monsters despite not needing to ever hit refresh.

@Clackers I am thinking that this might not be a good idea. This last All Rare war is an example. One of my battles must have gone 6 rounds, I think. I recharged every monster at least once. That said, though most battles don't last that long at least not for me. What if you could give such a monster extra moves? That might abuse the benefit, but maybe not so much. Looking forward to hearing what other pros and cons there are.

one of the many selling points of nemesis is those 0 stamina skills that can give regenerate and counter thetys stamina devour so i think it is not likely that Sp will put them on other monsters