You would be much better off not using the inverter to charge the RV battery. I lived in my motorhome for years without turning on my charger. I have a 415 rated watt solar system.
You could install the solar panels on the shed, then install one battery either there, or in the RV, and run the wiring to it. 12 volt wiring can run between the two batteries and charge it fine. #10 UV rated direct burial romex will work fine, you can buy it by the foot or in 25 - 50 foot rolls at Home Depot, Lowers, ect.
#10 will carry 30 amps, but for the most part, your RV power load will stay under 10 amps on average, so voltage drop should be insignificant. By leaving it at 12 volts, you will save a lot of power that might otherwise be used by the inverter and charger.
Say you had a 10 amp load in the trailer at 12 volts or about 120 watts of lights on. The charger would use about 150 watts to make that power. To supply that power from a inverter, then the inverter would be consuming about 180 watts - perhaps as much as 220 watts, depending on it's efficiency. So your inverter might be drawing 20 amps to put 10 amps into the trailer battery. Not what you want to happen.
I would really recommend putting both batteries as close together as possible, then run the solar wire from the panels, through #10 wire for each 200 watts, to the battery bank.
If you have a MPPT controller, and can wire all the panels in series so that it has say 80 volts input and 12 volt output (there are many controllers that will accept up to say 150 volts input and have 12 or 24 volt output, your choice) that would be ideal, but a MPPT controller is about 3 times as expensive as a PWM controller, and you might only collect another 50 watts per day using the much more expensive controller.
Panel prices are so low these days, you could find a 140 watt 12 volt panel new for $229, install that and a $20 20 amp PWM controller and get more energy than a $300 MPPT controller will get you. . .
SunElec.com has both controllers and panels for great prices. So if you upgrade, think about them. For RV's you want something with a aluminum frame. For homes, or your shed, frameless will work, but frames are easier to mount to.
Good luck,
Fred.
