No. You can only get the same look by standing in the same place. Otherwise, perspective will be altered. Nothing you can do with the lens (or camera) can get around that.

On a DX format camera, to get an "85mm look" — the look of an 85mm lens on a "full frame" 35mm-format camera, or FX in Nikon terms — you need a lens that's 1.5× shorter to match field of view, and also approximately with an aperture 1.5× faster to get the same approximate depth of field.

You can also use the "Brenizer method" of panoramic stitching to simulate a wider and faster lens than you have, but you'll still have to stand in the same place to get the same look.

So:

I know a 58mm FX lens (87mm DX equiv), shot from the exact same distance with a DX camera, will look almost identical and yield a very close perspective background.

Yeah.

But is the only other way to get the same "85mm look" on a DX camera, with an 85mm FX lens, to simply move backwards?

Nope. This will give you something like a "135mm look" (give or take), because that's what you'd use on FX to get the same framing from that spot.

Take a look at some of the graphics in the answers to What does it really mean that telephoto lenses "flatten" scenes? — I think they'll really help.