My friend has a cheap 1100W that sounds about the same.
A very low battery will stall the motor if direct connected to the 12V receptacle.
He uses a cheap AC battery charger to bring the battery voltage up a little, then charges from the 12V receptacle. The charging amps are then lower, freeing up power so the 120VAC receptacles can be used for other things while the battery is charging, like charging other batteries.
A very low ohm, high power resistor in the line may work. Or extra long wires to the battery.
A 220V 100W light bulb in line (in series between the 2 positives) may work, then remove the bulb and connect them directly after the battery voltage has risen.
The 12V receptacle probably makes around 14V, maybe a little more.
The '12V' receptacle is probably designed to simulate the voltage of running car alternator, which is also charging the car's battery.
I would not worry about overcharge, it would take a very long time.
In North America the center is positive. The outer ring is negative.
It would be connected to the battery the same way. Pos to pos, neg to neg.
A car battery needing charged is probably below 11.5V. The 12V recptacle will try to bring the battery to 14V instantly, and it takes so much power the motor stalls.
Resistance in the charging cables slows the amps so the motor can keep going and the battery keeps charging.
When the battery first reaches 13.5V it is still not full charged. It is more "surface charged". After disconnecting the charger the battery voltage will fall fast, without use.
To full charge the battery will take a long time at low amps, and a lot of gas. Not very efficient or cost effective ($ gasoline).
The high amps at the begining of charge is bad for the battery, but the lightbulb will help.
For a battery being used in a car it should be fine for sometimes use.
G-