It is very interesting! Does physical realization of the disk affects its behavior?The contracted length of the wire is equal to ##2\pi r##, yes. Thus when it slows down the wire will be longer than the rim of the disc and will fall off. This is true even assuming (unrealistically) that the disc does not undergo elastic expansion when spinning.
If you imagine making many radial cuts into the disc (so it looks like a jet turbine), you would find that these cuts open up as the disc accelerates, since each blade of the "turbine" would length contract in the tangential direction. If you don't make cuts, the whole rim of the disc must "want" to length contract - but is unable to do so due to the symmetry of the situation. So you will find elastic stresses in the tangential direction.
Presumably a spinning hoop would try to decrease its radius due to Lorentz contraction (massively overwhelmed by elastic expansion and outright disintegration in practice, I should imagine).
The spool can be made in the form of a ring without an axle and spokes. Will the wire stretch after stop and the rim will not?
What if the disc was winding a wire for many years and is now actually a spiral of Archimedes? What will happen after the stop? Which layers will separate from the axle and which will not?
Let’s assume that the spool does not contract in radial direction and the very long wire has been wound around the spool. The spool moves with velocity ##v=R\Omega## in the frame of unwound wire.
In the frame of the spool the rim of the spool cannot move faster than ##c##, the angular velocity ##\Omega## of the spool cannot be higher that ##c/R##, if linear velocity of the rim approaches ##c##
In the frame of the unwound wire, as speed of the spool approaches ##c##, rotation should generally cease due to time dilation.
But the wire cannot be whole in one frame of reference and torn in another.