Shouldn't temporada also work here?
Maybe, depending on the context. But as a rule, temporada is any season except the seasons of the year.
Temporada is s season within a season. Meaning: there are 4 estaçoes do ano, but within those estaçoes you might have hunting temporada.
What's wrong with época?
Epoca is more like an era
It's funny: I wrote this so long ago that seeing it again now, I'm thinking, "Wow, did I ever think that?!"