James
Which model do you have. Franks doesn't have a 3 way cam. It's part of the lens / helical focusing mount like a Leica M lens.
I've had two Linhofs that had RFs. My Master Technika that I used for nearly thirty thirty years had individual cams for each lens. My Technika 70 had a circular cam with 3 sides each corresponding to a specific focal length lens. The cam fit under the focusing rails and could be rotated with a button of sorts to position the correct side / cam for the appropriate lens. Each cam Was ground for a specific lens as as you say part of the serial number or focal length appears on the appropriate cam.
Both the Master 4x5 and Technika 70 2x3 have bellows. Franks camera has no bellows. It's a rigid body with a helical focus mount. Does your camera have a bellows? Post a photo of your camera.
I know from a couple of other Technika owners that I knew that the cams are close enough in many cases even if its for another lens of the same FL. I bought my Master just after they came out in the mid 70's. I had several lenses I sent to Linhof to get cams cut for. My 210 was a convertible Symmar and several years later I upgraded to a 210 Symmar S. I reset my infinity stops to match the FL of the lens (very important to set correctly) and tested the new lens with the old cam and it worked perfectly. I used this combination for fifteen or twenty years in my work without issue.
Give it a try. Just make sure the infinity stops on the rails are set correctly. Correct is when the focus is set to infinity and the lens is focusing at its best at infinity. I had several colleagues with Technika V that had cams from different lenses that worked properly.