Measure the voltage on the control trace, relative to battery negative, with normal power applied.
Then lift the lower end of the resistor that feeds that trace (above the red circle) and connect the now free end of it to the opposite battery wire; eg. if the control is near whatever voltage the battery should be, connect the resistor to negative, if near 0V connect it to positive.
That should hopefully turn on the lamp inverter.
Be extremely careful doing anything with power connected. The best thing is to connect an extra wire to the control track at one of the component joints and use that and the battery negative twisted / taped to your multimeter leads.
Or just connect the resistor end to each battery terminal in turn and see if either polarity works to switch the lamp on...
It is possible the battery charger part cannot provide enough power to run the lamp, without a battery connected; it could just be a fairly slow charger.