I'm going to quote bits of the question out of sequence.
Would the wave lengths change due to gravitational blue/red shift (due to the artificial gravity nature of a spinning disc)?
The frequency of a light beam moving down or up in a gravitational field doesn't change. Imagine you have a laser at the top of a tower, shining down. You measure the frequency at the top, and you measure the frequency at the bottom. The latter is greater than the former, but that's because your clocks are running slower when you're lower, because of gravitational time dilation. The light hasn't changed, you have. You can check this out for yourself by imagining you send a 511keV photon into a black hole. Conservation of energy applies. The black hole mass increase is 511keV/c². The descending photon didn't gain any energy. Since E=hf it didn't increase in frequency either.
or would this all be tied to a doppler shift?
Imagine we're in gravity-free space and I shine the laser at you. You measure the frequency, then you move towards me at some rapid speed, and measure the frequency again. You now measure a greater frequency. But again the light hasn't changed, instead you have.
If I were to spin a translucent disk so that the edge is spinning at .9c and shoot a laser beam at it perpendicular to the edge, what would happen to the light as it travels in one end of the disc and out the other?
We can probably simplify this by asking what happens to light if it's shone perpendicularly at a long glass block moving at 0.9c. And then we could liken that to you crossing a road. If the cars are stationary you cross the road no problem. But if the cars are moving at 90mph, you don't. So like anna v, I think the disk will be non transparent to light.
I would expect the light would become strongly redshifted as it enters the medium and then when it passes the center it would blue shift and escape the disk at the same wavelength it entered in.
I think you're looking at this wrong I'm afraid. Let's forget the above and say the light can go through the spinning glass disk. When it enters the medium the atoms/electrons are sweeping past it perpendicularly from left to right at 0.9c. As it approaches the centre this sweep diminishes to zero. As it departs the centre the sweep increases from zero, but going the other way. Then as the light exits the other edge the atoms/electrons are sweeping past it perpendicularly from right to left at 0.9c. But apologies if I've misunderstood the scenario here.