Answer: You will need to edit the MAPINFOW.PRJ file to modify the same projection in a different measurement system and adjust the False Eastings and Northings used for the difference.
For example below, the first line is the original projection expressed in meters. The second line is a copied modification of the first, where the units were changed from "7" (meters) to "8" (feet) indicating this coordinate system is now using feet, and the False Easting (2000000) and Northing (500000) components were divided by .3048 to convert them from meters to feet.
Example:
"California, Zone I (1983)\p26941", 3, 74, 7, -122, 39.3333333333, 40,
41.6666666667, 2000000, 500000
"California, Zone I FT (1983)\p26941", 3, 74, 8, -122, 39.3333333333, 40,
41.6666666667, 6561679.7, 164041.99