When you press on the clutch pedal a fork-like gadget presses on the throw-out bearing, which in turn pushes sort of in the center of a circular-plate looking gadget with a lot of spring-like arms on it, which causes the arms at the outer end to release the clutch disc from up against the flywheel where it is held by those springy-arms when the pedal isn't being pressed on. The clutch disc connects solidly to the input shaft of the transmission. So it is sort of like turning a pie plate upside down, pressing down in the center, and the spring effect of the metal pie plate causes the outer edges lift up a little.
That all is not happening correctly in your Mustang for some reason. The pie plat edges aren't lifting up high enough, which means the clutch disc remains pressed against the flywheel too strongly, preventing you from shifting when the engine is running, or when you start in gear with the clutch pedal pressed, the car moves forward.
So in my pea-brain way of thinking, the problem pretty much has to be from among -- in terms of the pie plate analogy --
- the center of the pie plate isn't being pressed hard enough, or in the right place; or
- the pie plate isn't doing what it is supposed to do when it is pressed on; i.e. lift up the outer edges.
The latter problem happened in a recent episode ( few months ago) in "Hot Rod" magazine's "Hot Rod to the Rescue" monthly feature, so you might take a look at that. In that case the springy arms just weren't springing up enough at the outer edge. The repair shop had to bend them a little to make it work correctly.
To the extent you have a custom build going on, which I think you mention may be the case, you'll probably have to take things apart enough to verify each of the steps above is working correctly. That may mean you have to remove the transmission and clutch and bench test the works until you are certain it works.