The stock 12v battery runs out first because it has less AH (ampere hours) than the Marine Battery. Then the Marine Battery in addition to trying to supply the motor will also try to supply the stock 12v battery which will deplete it sooner.
Running them in series will produce 24v to the motor and no doubt speed up the Power Wheels car. But those motors are small and eventually it could overheat the motor and cause it to fail.
Try running the car on just the Marine Battery alone or parallel in another stock 12v battery.
It'll take longer to charge but nearly double the riding time and the motor can take it. This is what I did on one of my kids riding toys years ago.
Lighter batteries? Sure, Lithium batteries are lighter and at 3.2v each you'd need 4 of them in series. Be prepared for "Sticker Shock" tho, Lithium ain't cheap!