It is probably safe to assume that the board was made by some anonymous contract electronics manufacturer and so the chance of getting any detailed info is quite low.
Two possibilities occur to me:
1 Put the thing back to how it was for 6 volt operation and use a regulator to reduce the 12 volt supply to 6 volts. This will be rather inefficient, you will waste as much energy in the regulator as is used by the light.
But it will certainly work.
2 Dismantle the transformer and count the number of turns on the primary winding.
Rewind the transformer with twice the number of primary turns, re-assemble it, pray to a god of your choice as you connect the power and see what happens.
If the primary winding is underneath the secondary, which is quite usual, you will obviously have to remove the secondary as well. A lot of winding could be involved!
JimB