Vibs
The only divine act that Rama performed prior to meeting Parashuram was reviving Ahalya! One has got to admit that this was a feat unachievable by normal humans: therefore, Parashurama's avatar must have ended long before, maybe when Rishi Kashyap asked him to end his genocide of Kshatriyas. Also, in 1-15-18:21, Brahma clearly asks Vishnu to take birth as humans. Vishnu in response tells them that he will reside in the human world and rule it for 11000 years 1-15-27:30.
Your other explanation about why the devas would want Rama back despite Vishnu being in Vaikuntha does sound credible. I'm however not sure that I'm totally convinced. If Vishnu could clone himself in different places and make everyone happy, who would have objected, and more importanly, would such an objection have been sustained?