The transfer switch will select between two inputs. It can't select between three inputs.
If you don't have a generator, you could certainly wire the inverter to a transfer switch installed as a generator readiness package in lieu of a generator and power the entire RV off of it (well, have everything live--of course you are limited in what you can power in total with the inverter). You would have to make sure you manage your loads properly for it to work well: shut off the converter (which may mean disabling some other things if they happen to share a breaker), not use the air conditioner or electric water heater element if you have one, perhaps force the fridge to propane mode, etc.
It's sometimes simpler to have a transfer switch somewhere after the main panel and only power certain circuits with the inverter, obviously just those that supply power to things you want to use and not those for the air conditioner etc.
If you have a generator, you could wire the inverter to a socket into which you can plug the shore power cord when using inverter power, which is effectively the same as using a transfer switch, just a bit less automatic. If the things you want to use from inverter power are generally modest in power consumption, you could also wire those circuits solely to the inverter and rely on your battery charger to supply the inverter with power when connected to shore power or using the generator. It would potentially slow down the battery charging some. That can work especially well for things like satellite receivers that have a comparatively long start-up cycle after a power interruption before they relearn the channels and program guide and similar things, as it avoids power glitches.
