I’m not sure about that - the converter will take the 120v your inverter is supplying and convert it to 12v to supply your 12v stuff AND charge your batteries. But your MPPT controller will already be doing that with power from your solar.
So theoretically yes, and at a minimum it is not needed and it’s probably inefficient. My house batteries were 24v (actually a 32.8v lithium bank) and my inverter was 24v (actually 28v) not 12v, so I always ran the converter off inverter power like any other 120v appliance/equipment to supply 12v, because I didnt have 12v house batteries. The systems were separate.
Since your solar and house systems will both be 12v, this is something to sort out with someone who has 12v solar experience. My guess is that you wont need your converter but it wont hurt anything leaving it on, and when there is no demand on it because your batteries are charged from solar, it will sit idle. But it might waste power, and it will certainly be unnecessary, so your instincts to shut it off are probably correct.