Cut Carriage Flex Cable
The symptom that you are describing is very common with aging SP-300/V and SP-540V printers. It's most likely a bad cut carriage flex cable. This is the cable that connects the cut carriage to the main board. It's sandwiched between two clear Mylar guides.
Replace the cable and (2) guides and your problems will go away. (Less than $50 parts / 1 hr or less labor)
No need to replace the linear scale... it has nothing to do with measuring media width. Media width is determined by reading the position of the two outer pinchwheel sensors. The pinchwheel sensor is on the rear of the cut carriage. The printer determines the location of the cut carriage using the rotary encoder in the scan servo motor. The linear encoder is used to determine the location of the print carriage... and doesn't play here.
Not a tricky repair, but you might contact someone experienced in this type of repair as you do have to remove the delicate linear scale to properly clean the track and replace the cables.
PM if you need assistance.
