I think it'd be a great addition to the housing part of the game. I see it being a job the could be utilized by all crafters based on what is being mounted.
Primarily I thought this up as a way to make legendary fish useful and the idea evolved from there.
Basically allow carpenters to make the bases, and the other crafting classes make the beasts based on their "make-up". For example rocky monsters to armour and black smith, fish to apothecary/ culinarian, vegetation based monster to weaver, and so on.
As far as the monsters themselves I'd say make it a very rare drop that when killing something the player obtains a " carcass" or something.