I managed to have my stamp transferred at a different immigration office, but I had to contact the office where the stamp was issued to have them "update the stamp in the online system". The people at the old immigration office that I talked to over the phone had no idea what I wanted done, but eventually I got an immigration officer at my new office to talk to them, and they sorted it out.
Not sure if that's something Phuket/Trang will do, but it seems to be doable in general.