I have run both types and mostly the Lithium for the past few years. areovoltz is the one kitfox put in mine I think. It works fine if you don't run it down too far . They do have their own charger that is complicated to use. I also use a lithium maintainer / charger than works fine also. the answer to your question is no you do not need to keep a charger on it when there is no load.
On my Remos I recently traded the odyssey for An Antigravity lithium battery with more power and saved 12 lbs. They are much less than the earthx and it fit in my existing box. Google them.