I can give some information about this as regards to beavers in the medieval period and provide sources for you. Firstly for the meat to be licit during lent it just needs to be thought of as fish - not declared to be so. Whilst it is not beyond the realms of possibility that there is a piece of Canon Law specifically regarding this (after all there was a statute forbidding men to be scared of animals whilst obliging animals to fear men!) I think it is somewhat unlikely and unnecessary. If monks or priests thought that Beaver tails had the nature of fish then they were allowed to eat them even without papal blessing. The idea that beaver tails did have the nature of fish goes back to Pliny the Elder and his natural history and is repeated by authors throughout the Middle Ages including Gerald of Wales and Bartholomew the Englishman, it is also found in bestiaries (although frustratingly not in the Aberdeen Bestiary which is a great online source I recommend in general). Basically with all this authority behind it this is not a claim which needed much Papal pushing - and it is best not to think of the Pope as having comprehensive influence on the day to day lives of everyone in the church anyway. As to why it was seen as fish-like one reason given was that it was scaly like a fish rather than smooth like skin or hairy, that it is used for swimming can't have hurt this categorisation either.

Moreover the Beaver tail exception is not even the most bizarre dietary exemption or quality of the Beaver. Gerald of Wales describes Barnacle Geese being eaten by fasting Irishmen due to the fact that they (supposedly) hatched from barnacles! (obviously whales, dolphins etc. were also seen as fish) The beaver, meanwhile, is depicted castrating itself (the latin for beaver is castor) when pursued by hunters after its testicles, thought to be medicinal, and cheerfully throw them at their pursuers - if new hunters came beavers would point to the absent patch where their genitals used to be. As for the capybara I am afraid that this is outside my period but I hope this partial answer goes some way to help. The places you will find what I have talked about are: the History and Topography of Ireland by Gerald of Wales, the De proprietatibus rerum by Bartholomew the Englishman (despite the Latin title you can find a translation by John Trevisa, a bit old), The Natural History by Pliny the Elder.