What are the characteristics of light entering a disk spinning at near $c$?

Light is an emergent phenomenon from the confluence of innumerable individual photons with the energy $h\nu$, where $\nu$ is the frequency of the emergent macroscopically beam.

The question then reduces imo to "does a photon see/interact with a rotating "transparent" disk the same way it does with a gravitational field".

Checking on the special relativity part first: What does transparency mean to a single photon?

A disk rotating at such speeds will no longer be transparent to the photon, imo, as the probability of finding an available quantum state will be very high (depending on the radius of the disk) but the quantum states of the molecules composing the disc will also be stressed to new quantum mechanical solutions.

I agree with this exposition about the behavior/transformation of solid bodies at relativistic speeds.

So the disk will be non transparent to light, as it is for photons, just from the effects of special relativity.

As far as general relativity goes the main question is, "does the accelerated motion of the disk distort space time within it in a graduated manner"?.

The answer should be yes, in a graduated manner, as all predictions of the mathematics of General Relativity have been validated up to now. It can not be tested with photons though even in gedanken experiments. One could try to think of one with neutrinos, but not in the set up described.