Not sure I understand what you are proposing, but my gut says no.
What is the "control switch input"? The RV auto transfer switches I'm familiar with don't have separate control inputs. Instead they sense which of the two power sources has adequate live voltage and switch to that one. If both sources have power, one is given priority over the other. in an RV, usually the generator side has priority over shore power.
Please explain what is currently wired to that auto transfer switch (ATS), e.g. is it the typical shore power cord plus a generator? And is it a 30A/120v ATS or a 50A/240v ATS?
Also, does the Sungold 4000 have a charger built-in? Perhaps
this model? If so, you can eliminate the existing converter/charger and let the new inverter handle charging as well as inverting.
Are you expecting everything in the RV to work from the inverter the same as it does from shore or genset power? That's not going to happen for practical reasons, but it can be wired such that it could work that way if you have enough battery power available to meet demand. A 4000W inverter draws around 350A at full load, so you need a huge battery bank to come even close to that amount for more than a few seconds.