You won't convert to a 6 volt system although you can convert to 6 volt batteries. Two 6 volt batteries in series provide 12 volts at the same AmpHour rating as one of the batteries. Two 12 volt batteries in parallel provide 12 volts at double the AmpHour rating of one battery. Four 6 volt batteries in series/parallel provide 12 volts at double the AmpHour rating of one battery. Your TT operates at 12 volts. If you have abused your existing 12 volt battery (the word "dead" indicates abuse) you will have to replace it if you want to add an additional battery, even if it was the same as the one you have now. How much room do you have for batteries? Is that area isolated from the TT interior and vented to the outside?
I would see how much room you have and get a pair of the biggest 6 volt true deep cycle batteries you can find and fit. 220 AH (two 6 volt batteries) should be enough for your described usage, although what do you have or plan for a charging regimen? Four batteries would give you more than enough power for the use you describe.