I don't have a Geany installed as system so I can't try using the menu, but from the command line it follows the working directory...
I usually run Geany from a shortcut. Running from command-line does follows PWD.
If the remembered save-as folder is just put in the right place, along with checks to account for projects, shouldn't the existing fallback behavior pretty much remain intact? (So instead of "home", I should have referred to "the rest of the existing fallback".)