I think you misunderstand what a 6-volt system is.....it's two 6-volt batteries that are wired in Series, to create 12VDC. Because of the larger size plates and such in the 6-V battery, you will get a longer life from it, but you could also just wire two 12V batteries in Parallel, which would double your service life from them.
Try to keep the batteries about the same size and age....or the smaller, older (weaker) battery will drain down, and be useless.
Depending on the age of the 12V battery you have, and if you could use it for something else, I'd probably go for a pair of 6V batteries in it's place.
But, there are also some folks, who feel that you should stick with 12V batteries, because, if you have a failure of one of those 6V batteries, you are out of business....at least with just one 12V battery, you can still limp home, or stay another day!:)
Read up on how batteries can be wired together....parallel vs. Series, so that you understand the operation before you do anything....or ask a friend who knows these things, to help you.
On edit, I guess we were typing at the same time:)
Anyway, if you have the 6V batteries, you don't have to worry about your inverter...you would just wire it up to the both batteries to get the 12VDC connection from them (with batteries wired in Series).
Hope that helps.
In your current situation, I'd just get another 12V battery that is the same size as what you currently have, and wire it in parallel.:)
Good Luck:)