If you are planning on installing a solid, 3/4" thick wood floor, you need a 3/4" wood subfloor MINIMUM as a nailing base. The subfloor can be floated, glued, screwed and nailed with a moisture barrier underneath it. If the clearance between the slab is and the bottom of the door is less than 1&3/4", you will have problems. As it is, even at 1&3/4" clearance, you will still barely clear the proposed floor and not have room for an area rug. The easiest way to resolve is your idea. Have the builder re-hang the door. He may not like it but it is doable.
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