It does indeed appear like it would be possible to add module effects from one starbase type to another - but in practice the program does not process them. For example, giving "productionassist" ability to a mod on a military starbase yields no benefits - the Details tab on the colony pages lists no outside bonus affecting the colonies covered, and the actual production values do not appear to change either. It is much harder to test the functionality of other module types, eg trade modules, culture, however I can't see why they would fare any better.
Well crap. I thought it did work. Saw the ability listed on the Starbase Report, but never bothered to think to actually check if they worked. There goes one idea out the window. Oh well, we can still hope that maybe the devs will be kind enough to throw modders a bone on this matter.

re: influence expansion - I didn't know there was already a function in the game to do this. Could you tell me more?
This lists a StarbaseRange ability, although I can't seem to get it to work. Also, I just realized the description of the ability seems to indicate it increases ship range, but I swore I saw a mod that had a module that increases Starbase range... have to dig around and see if I can't find it again.
Edit: Found it.
TOV's Starbase and United Planets ModForum Thread