My history with TT batteries is mixed but they usually last about 3 years with some care. Even the so-called maintenance free marine batteries need to have their water level checked regardless of advertising hoopla. If you are always hooked to shore power, almost any brand or size deep cycle battery will work just fine with a Group 24 kinda being the 'standard' size though small in amp hour capacity as well as physical size. Our pretty big 5th wheel had a Gr 24 installed by the dealer and I could barely complete a setup without it running down. That's been replaced by a Gr29 from Walmart.
Manufacturers/brand names in the US (Lead-Acid Battery Manufacturers and Brand Names List 2016
) - many are made by Johnson Controls, even the Walmart brands and the very expensive Optima AGMs. Interstate is another large manufacturer with dozens of brand names.
In 40+ years of 'playing' with RVs, only one converter has failed - and that was only an intermittent problem that a good kick usually fixed until the next time. That particular converter was installed next to the breaker box and under the fridge. I've forgotten the cost - but it solved that problem.
Oh and more reasons the lights dim - corrosion at a battery terminal end connection or oxidation or loose connection on the ground to trailer frame or breaker box.