A new rear cross member would be a weld in bit... There is no guarantee that the holes in it would be lined up any closer to their stock position than where the bumper is sitting now.
The geometry is clearly off a bit somewhere. The possibilities:
The bumper is mounted too high.
The tailgate is warped/mounted too low (unlikely).
The bumper is mounted too far back.
If you were bolting on through a stock rear cross member, the old hitch holes would help position the bumper vertically and pull it toward the front of the truck. Those holes aren't there (cut away from cross member), so the bumper is moving to whatever position the other mounting hardware pulls it to.
At this point I'm assuming you're keeping the bumper and that it will get more or less permanently mounted on the truck.
So... IF all of the other mounting hardware were loosened up or removed - including the carriers (they'd only get in the way for this) - can the bumper be put in a position over the old cross member where the tailgate can clear it? I.e. - is what you're after possible?
If so - then maybe find that position, drill 2-4 new holes 6" or so left and right of the old hitch holes through the bumper and the cross member (there should be metal there). Then you can snug it up, test the tailgate clearance, then bolt on the rest of the mounting hardware.
If you're dead set on replacing that cross member - I'm sure it can be done, but I doubt it could be done to the mm+/- in 3 dimensions that you would need it to be done in for stock-like fitment.
Hmmm...