I have a compressor with basically that same tank and what looks like exactly that same hoist, and this move was no problem. The way I see it, there are two ways to do this, one tedious but safer and another quick and ********-clenching.
Tedious but safe: If you still have that pallet jack, pull the compressor out, jack it up as high as the pallet jack will go, and stack enough boards under the pallet that you can slide that engine hoist under and lift. I've seen people on YouTube lift 10,000 pound CNC mills up high enough to pull a trailer underneath just by sequentially tilting up corners a half-inch and stacking shim boards one at a time. In your case, I bet for such a low lift and relatively light piece of equipment, this wouldn't take but an hour or two.
Fast and probably still safe: Drag the compressor out from that corner and get the hoist as close as possible, with the feet of the hoist right up against the pallet and the hoist aligned with the compressor. Make sure the legs of the hoist are extended farther than the arm so the hoist doesn't tip over forward when you start leaning the compressor over. Run a lifting strap under the pump platform and to the hoist (as pictured below), unbolt the receiver from the pallet, start lifting. The compressor will start tilting and dragging across the pallet toward the hoist, eventually picking up off the pallet and dangling. Move pallet out the way, set compressor down, reposition straps for better balance.
If you really want to add safety to this last method, you could use straps to anchor the hoist to something behind it so it can't tip over.
A 700 lb. compressor is nothing to a hoist like that. Just, as suggested before, grab from the top, as the assembly is top-heavy.