After looking more closely at the greenhouse frame, we've decided to leave the metal studs raw. It is not unsightly, and will be covered by trim on the outside.
My biggest concern is mildew, which is an ongoing problem with the white paint on our house. The trim on this house was washed down with bleach in August and is already graying.
And while the first paint job would be easy enough, if costly, others down the years would be tedious.
Best to leave it alone.