My experience with 6v systems is that if the battery is 100%, then 6v can be as good as 12v. But this assumes that there are zero unneeded resistances anywhere in the system, and no extra loads.
When the lights are low and starting is iffy, the actual losses that you want to address may be distributed over many components, each one just a few percent of a volt, but they all add up. As with any research or investigation exercise, you can only decide what to do on the basis of the information you have. The better the information, the better the chances of a good result. In the quest to chase down the missing volts, the main tool is a volt meter that can give you a reliable reading to better than one tenth of a volt. This sort of meter is really cheap from any electronic hobby shop or chain store. Then, you just start out measuring voltages in a systematic fashion. I suggest you do this by connecting the -ve wire of your volt meter to the ground terminal of the battery with a 8ft length of wire. Then measure and record the reading you get on the positive terminal of the battery. Then turn on the lights and check it again. Should be very little change. Then repeat this process on every access point you can on the electrical circuit between the battery and the lights themselves. Also check the earth connections from the lights and on the body where the earth connections are made (these reading should be dead zero with the lights on). Use the wiring diagram to make sure you are covering everything along the path. From the result of this exercise you will be able to identify where volts are being lost. It may be in terminal connections (you can just clean these up), but it may also be within components (e.g. dip switch) which may need to be replaced. It may be along a length of wire( replace with a heavier grade or duplcate). Suspect everything.
After addressing all the problem areas that you have found, repeat the check at critical points.
A simpler cruder check is to leave the car standing with the engine off and the lights on for a few minutes on a cold day, then feel the main lighting related electrical connections and components. If they are warm... this is probably the biggest part of your problem. But maybe not all of it.
6v halogens can give an excelent light.
Cheers, Brian