Removed exterior door panel, took out the soap dispenser and then discovered that the inside of the door had been bent in, apparently during manufacturing. Banged it out, checked it with a straightedge, and reinstalled the new soap dispenser with a little silicon for the slight irregularities in the door. Worked perfectly. Uses torx screws.
The real story here was that I ordered the part on Wednesday afternoon, and it was at my door before noon on Thursday. Short of a Star Trek transporter, I don't know how they did it!