Your converter/charger does a lot more than just "charge the battery when on shore power"... It provides all the 12 volt power for lights, refrigerator/water heater/furnace operation as well as all the "phantom drain devices" such as remote control systems, LPG/CO monitor safety systems and charges the battery with what's left over.
Most converter/chargers are WFCO and typically are rated at 55 amps continuous output. Some inverters have the capacity to provide the entire output that equals the WFCO converter, some inverters have a 10 or 15 amp "battery charge system" as an add on when the inverter senses shore power.
Depending on the rating (continuous or intermittent) and the maximum continuous DC amperage output, you may be able to replace your converter/charger and just turn the circuit breaker off, shutting it down.
On the other hand, depending on the capability of your inverter battery charging system, it may or may not be adequate to provide continuous power to your RV or it may/may not be anything more than a "small battery charger".....
If you have both the battery charger from the inverter AND the converter/charger operational at the same time, there's a very good chance that they will each "sense the output voltage of the other" and both will think the battery is fully charged and shut down any charging to the battery. So, do some "due diligence" on which brand and model inverter you're considering. It may or may not even be capable of supplying enough power to operate the trailer systems.
2015 F250 6.7l 4x4
2014 Cougar X Lite 27RKS