Feed to a barn for some lights, receptacles and an air compressor

I would use #2 aluminum wire, if in conduit use XHHW insulation, if direct buried use USE insulation. Need 3 #2 for the two hots and one neutral, and 1- #4 for the equipment ground.

Neutral needs to be insulated and isolated in the barn panel, and a separate equipment ground bar for grounds, and need at least one ground rod grounding the equipment ground bar.

Is #2 uminum the cheaper way to do? What about direct direct buried copper wire could I use # 4

Get quotes for both options.