I actually work as an engineer for a water softening company and can tell you that unless you're using an Ion Exchange resin (softener), there really is no practical way to remove the calcium carbonate. A softener basically trades the ions of sodium or potassium chloride for the ions of the calcium. Only by removing the calcium carbonate and calcium sulfate can you truly have soft water.
The good news is that there are several very small units sold specifically for RV's that can be hooked up when you connect to whatever campsite you're parked at (assuming you're in one with services) You can also fill your tanks with the treated water before you leave.
These smaller units do not backflush automatically, but require that you clean them manually by drawing a saltwater brine into them (actually pretty easy using a bucket and rock salt you buy at home depot etc.) So every couple of hundred gallons or so, you'd have to clean the unit.
If you do a search for "RV water softener" you can find several very reasonably priced units.