it is entirely possible that the brake controller could be adapted to run on a 6 volt system, most electronics run 5-5.5 VDC, so in theory it would operate if it uses a regulator. If it uses a resistor, it would require some alterations. the controller would also have a fast and slow gain adjustment on it to determine how much to apply the brakes on the trailer.
The brakes themselves would be the object to question, theoretically one could make them operate on 6 volts, but I need to do some testing with voltages and strength. There were older 6v systems, and I think you could make them operate with the modified brake controller to act as a modern setup. I will do some looking into magnets and voltage when I get back home tonight.
I am sure you have heard this many times, but other than nostalgia, why not convert to 12 volt? If you are going to haul a trailer, then I would recommend 12V. It would be cheaper in the long run, and likely safer. If the brake controller goes out, that would be special, brakes may be special, you couldn't haul the trailer with a different vehicle due to voltages. accessories would be few and far between, light bulbs would be more difficult. If you are going camping, you would likely want an inverter to do things such as charge phones, laptops, anything 110, and I am yet to see a 6V inverter or phone charger. just food for thought...