Adjustable-height, side-mount, heavy duty casters? Need to move a heavy air compressor
What about thinking above instead of to the side? Attach the large casters to a hefty beam that extends wider than the base. The casters would have to be taller than the current top of the base, or spacers added to the beam to make it thus.

The beam then hovers above the base, one beam per side, with long bolts, perhaps lag bolts extending through the beam. As you tighten the bolts into the base, it is lifted from the floor.

This is a suggestion for a temporary transport, although it could be modified or left as-is for permanent movement options.

EDIT: The reference to ten inches of air brought to my alleged mind a product known as an airbag lift:
Amazon link to airbag lift

The text references five hundred pound capacity (per bag) and a 3/32" thickness. You'd have to use a bar to get that space in there, but it's less than an eighth of an inch and certainly doable. Three inch lift capacity means you might have to step-lift your pallet, but you'd eventually get sufficient height to attach ordinary casters within your existing footprint. US$28 isn't particularly expensive.