Both methods have advantages. Using the converter/charger is much EASIER as you can also use the 120 volt power for other things while charging. Of course you CAN use it while using your stand alone also but many smart chargers read the battery state of charge to determine the charge rate to be used and if you're converter is also charging the battery state would appear to be fully charged, setting your stand alone to it's minimal charge rate. The fix for this is to disconnect the negative lead while charging. Ultimately it depends on how good your stand alone is compared to how good your converter/charger is and whether you take into account convenience.
Good luck / Skip