I don't know whether you''ve already started your trip, but the tips might come in handy for somebody else anyway.
You are required to have a fishing license, which I believe isn't too expensive. You can probably get one at any gas station or fishing shop around the area. Fly fishing is allowed almost everywhere. Catch and release is mandatory in most places.
You can have a look at the
fishing regulations. I believe there's a link to the full updated text.
I don't know if finding fly fishing equipment rental will be easy. I haven't been able to find any rentals, at least in Esquel and Trevelin (two cities 350km south of Bariloche).
I've been writing a short article for a friend on the most prominent fly fishing places in Patagonia, you might want to take a look at it before planning your trip. I actually came across your post looking for any places I might have forgotten about, so any comments are obviously welcome.
Link:
Fly fishing in Patagonia
Good luck with your trip!