Light Banding When Photographing Negatives. Light Source Cycle/Flicker Issue???

I have been photographing negatives for a long time using a light table.  I recently moved into mirrorless and this time I got serious banding.  The amount of bands changes depending on shutter speed.  The attached images was shot using the Sony a7RV and 90mm 2.8 Macro.  The exposure here is 1/30th @ f/5.6.  The other photo is the setup.  I use ABS tubes cut to a specific length for a given negative size which eliminates outside light and is spot on for filling the frame each time.  The light table is old (from the 1980s) and used daylight balanced fluorescent lights.  I am wondering if this is a light cycle/flicker issue of some sort.  If I shoot much slower, 1/4th or so, the banding is much less noticeable.  However, then I have issues with motion.  QUESTIONS:  1) Am I correct that the light source is the issue?  2) Can someone recommend a better light source?

Thanks for any help with this!

Do you see banding when viewing the image data at 1:1 (100%) zoom in, say, Photoshop? That is very important to consider. 
Also keep in mind that if you do see such banding at that ratio, it may be in the actual image data or it could be in the display preview path. But let's start with viewing the image data ideally. You might consider uploading a sample to something like Dropbox so others can examine this. 

1 hour ago, digitaldog said:

Do you see banding when viewing the image data at 1:1 (100%) zoom in, say, Photoshop? That is very important to consider. 
Also keep in mind that if you do see such banding at that ratio, it may be in the actual image data or it could be in the display preview path. But let's start with viewing the image data ideally. You might consider uploading a sample to something like Dropbox so others can examine this. 

Yes, the banding is still there at 100%.  As I mentioned, it changes depending on shutter speed.  My guess is that if I go below 1/4th second it would be gone but that has been making crisp copies impossible.

If you don't see banding at certain shutter speeds, that's the issue. Likely your shutter speed is not synced up with the refresh rate/frequency of the light source. 

As for a better lightsource, I've had success with the Kaiser Slimlite Plano LED light box:

https://kaiserfotous.com/search.php?search_query_adv=slimlite

https://kaiserfotous.com/?gad=1&gclid=CjwKCAjwkNOpBhBEEiwAb3MvvZZYZFiASikjA2QCGIHRNdi3MWaAX3-BOO4Nx4cPbhs_JyjON-sjuhoC8oUQAvD_BwE

Yes it's the fluorescent light. You'll need a frosted filament bulb or LED as digitaldog suggested. With the set up you have there now, if you remove the negative, you should see flickering on your camera's screen. Alternatively, hold the camera up to a bare fluorescent light bulb and you'll see more intense flickering.

I use flash for DSLR scanning. It has a well defined colour temperature, eliminates camera shake and gives much more flexibility in terms of aperture.

I also use flash. Excellent quality, no banding or slow shutter shake.