No, it wouldn't work very well, PATS keys needs to be inside the "loop"/wand around the key slot, this loop is just an antenna so is pretty much the same on all models.
When you turn on the key the loop sends an RF blast, this gives the RFID tag in the key just enough power to respond with its unique ID #, PATS checks # against it's list in memory.
If you put the Mountaineer key very close to the loop, inside cover, and for sure didn't use another PATS key in ignition, it might work.
You could remove the key cylinder and have it repinned for the Mountaineer keys, door keys would be different though, but they could be changed as well.
I think 2000 Ranger PATS still had the PATS module above the glove box, not sure about the Mountaineer.
PATS module and Computer(PCM) are "married" so must be used as a pair.
Only a Ford Dealer can "re-marry" PATS module and Computer if either needs to be replaced, and I believe it can only be done in the vehicle.