How to power AC bike lights with batteries?
This will probably work. Connect up a DC source to the lights and try it. Start at about 3 V then increase this as far as 6 V until you get the light you need. You need to aim at getting the output you need with the minimum of DC voltage.

The issue that concerns me is power dissipation in the rectifier. Using DC all of the power is going through one (or two) diodes instead of two (or four) diodes with AC. I can not be more precise without asking the manufacturer or having a circuit diagram.

If this works you will need a circuit to avoid overdischarge of your rechargeable batteries. This will disconnect the batteries when the battery voltage drops below their safe threshold. This is particularly important for Lithium chemistry batteries as it is possible to ruin the battery by overdischarging it.