Shortly after being hired, Rex got stuck in a sitting position and was unable to follow. I believe I recognize what was going on, from a similar problem Dogmeat had in Fallout 3.

in Idle Animations. in Creatures\Dog\Idleanims\DogCommonIdle\, the animation DogSitLooping, I think that you will want to add a condition to exclude Rex (or exclude mobs which have been set as player teammates) from that idle. Its a loop forever sitting idle and Rex seems to have cycled onto it, which means he was prepared to sit for a very long time.

In my game I got Rex moving by doing disable/enable on him.

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V22ElevatorDoorScript does its movetos of companions without checking for the companion having been set to waiting by the player. It only checks if they have been hired.