I don't see any problem doing all those things.
I doubt that charger would overcharge or harm the battery in any way. I would check the voltage it provides to the battery with a voltmeter. Anything up to 14.5 volts is safe as it is what engine charging provides. The current will automatically diminish to zero when the surface charge on the battery reaches the voltage the charger provides. If the voltage is higher than your converter provides it will charge faster and better. Old converters like mine only supply about 13.5 volts which takes forever to charge fully.