For starters don't make your new base a permanent fixture,
if it is removable, you don't have to account for it while mounting
the jack.

Given yor criteria for inside mount, I would look to mount it outside
on either the rear bumper or front one. Looking at your pic site,
I think figuring something out for the rear bumper is your best bet.
I don't think you can do much with the front, it sits in to close.

How much clearance do you have between the rear bumper and the
gate when it is down? You migh be able to weld some bolts to the top
and mount the jack there. Otherwise you might consider building a tire
carrier that the jack can mount to.