Break the mounting plaster and remount on a different articulator. It is done all the time. If it is already on an articulator why would you need to transfer it anyway?
Although this method sounds time consuming to me, if it is working for you, providing consistent, accurate, reliable, repeatable results, then go for it! "If its not broke, don't fix it!" Not all doctors are going to send their articulator with each case they send to the lab. In which case the model is being transferred from one articulator to another. Hopefully in such cases, you have an articulator that is compatible and calibrated with the doctors. Everyone in the industry has their favorite brand or model of articulator they like to use for a multitude of reasons. Maybe its the old articulator they used while in dental school, or they may have the latest, greatest, articulator on the market. Whatever the case, the lab may not have a compatible articulator that is calibrated to the doctor's articulator's settings.
The best advice I can give is, choosing an articulator system should be a combined effort between the clinician and the laboratory. Take your time when researching all of the systems available and, ultimately, choose the one(s) that will best suit your needs. The initial cost may seem expensive, but purchasing a quality articulator system is a worthwhile investment for producing high quality products. The clinician and technician should decide together which articulator to use for the restorative work they engage in. Staying within that articulator system will help provide consistent, accurate, and repeatable results!