Some companies do make a unit that is a combination Inverter and Converter, but this is not common and they are separate units.
Neither the Inverter or Converter are used to start your RVs generator. The COACH batteries are used for starting your generator. So, it sounds like you have two problems.
I suggest that you go back and recheck the wiring to your COACH batteries and make sure you have everything wired properly. Your Converter is used to keep your COACH batteries charged and your Inverter will use your 12-VDC from your COACH batteries to power a few receptacles that you use for your TVs and maybe for a computer while you are traveling.
Considering all of this, your symptoms suggest that you have not rewired the new batteries properly, so again, make sure that you have things hooked up properly.