It's not really the size, but the voicing. Some cabs/combos loaded with a 15" could work just fine, but the Fenders are voiced to suite electric bass. IME, that makes a double bass sounding artificial, electric and boomy (as the amps tend to emphasize lows and cut mids).
Having an amp that you can get a flat response out of is really helpful, even if you don't end up EQing flat.
If that is the amp you're working with and don't want to try something else, I'd suggest looking at a Fishman Full Circle pickup. They seem to do well with electric bass rigs, and can stand up to a relatively good amount of gain before feedback.
Best of luck in your search!