My initial thought was to do what I think you are attempting to do. Rewire the TT to run off of the batteries with no shore power and then use an external "car" battery charger to supply the 12 volts to keep the batteries charged.
I have a 3 stage external charger. It was about $100. I have never seen on e that was 4 stage. It seems to me that it would be better (less effort for one) to spend an extra $50 on the converter.
Unless, of course, you already own the external battery charger and can dedicate it to the TT.
Wayne in Coarsegold near Yosemite
2013 Forest River Solera 24R
2014 Jeep Wrangler Rubicon
ATV trailer for the following toys
2008 Yamaha WR250R,2009 Honda CRF150F
2008 Polaris Scrambler 500