Yes, both chargers can run at the same time and yes, you stagger the charging cables with a plus on one battery and the negative on the other battery.
One thing to note, some smart chargers are smart enough to sense a high voltage at the battery, caused by the converter doing it's charging, so they can erroneously assume the battery doesn't need much charging and dial back the charge current.
It's more work and more money, but replacing the charger in the OEM converter with a higher amperage, more modern and smarter charger might work better in the long run.
I picked up a Victron AC to DC 14-amp charger for a specific charging task last year and I love that it's super smart and has Bluetooth that works with the Victron smartphone app. They do make higher amperage AC to DC chargers, too.
https://www.amazon.com/gp/product/B07TJK3X71