I agree with most of what has already been posted. Your batteries are 4 years old and have one or more bad cells. Any battery wth a bad cell can be charged until Christmas but when a load is applied, in this case starting the genny they immediately go flat. A Hydrometer will confirm this.
The so called dual purpose batteries found in many RV's are more prone to this than true deep cycles. My 2012 nexus Class C came with two group 27 Interstate, pure deep cycles and they lasted 5 years. They showed weak at the end of their life when I tested with the Hydrometer but there were no bad cells - they were all equally low.
The converters on most RV's are between 35 and 55 amps {probably the case with your larger C}, and while they do in fact charge they are not dedicated chargers. The solution is to simply confirm my theory with a Hydrometer and then replace both with pure deep cycles. I got mine at Batteries Plus for $100 a piece.
I also replaced the engine battery as it was 5/6 years old {the chassis on my 2012 is in fact a 2011}, and should now be done buying batteries for the next 3 - 5 years which is about right given regular use and proper maintenance.
:C