It doesn't have jacks already on it? I thought all the Lippert framed units had jacks under them, maybe not.
Anyway, I would most certainly go with the scissor style "leveling" jack and not the "stabilizing jack" with the thing that goes coach wide and has an arm on either end that drops down. We've had some experiences with those stabilizing jacks (not on the Sunline though) were if the situation is on dirt/uneven ground, if you try to pick up the trailer whatsoever, the jack will be very hard to turn to do that. If you happen to be in a situation where you haven't locked all the tires and the tongue jack starts moving, this style jack will just bend like it's nothing. At least if the scissor jack starts sliding off whatever block you use or even if it's just on the ground, I think it should still hold the trailer up and not bend horribly.
Not to mention, the main force contact point for lifting on the stab jacks is way in from the frame. It does depend on the unit, but it could be up to like a foot in from the frame rail on each side. So that means that any lift you get will mostly be put on the small channel for the jack frame and not applied directly to the frame rail, so this will cause quite a bit of bounce inside the unit too.
Here's what has worked well for us for battery tenders. We've been using them for a few years now for all the stuff since a couple of the Sears ones we had started acting up. I know some have mentioned on here in the past that they like a charger with a desulfate mode too, but I'm not sure what brand has one. Regardless though, I would plan to take the battery off the trailer and put it either inside your garage or in your basement and put it on charge there. Also, make sure you get a 12V charger and not a 6V.
Battery Tender Plus - Batterytender.com
Jon