A small camera and reduction lens probably ends up cheaper than a 4/3" sensor.
Not sure about image quality of those lenses, ....
Yes, that seems to be the crux. There are companies that will cheerfully sell you a $35–45 dollar reduction lens, but the optical performance can be quite poor. When I was adding a camera to a different stereo microscope, good reduction lenses seemed to cost around 600-900 dollars/euros. Granted, they may have been marked up 100 or 200%, coming from low volume, Western vendors. But I could not find anything decent in between.
That persuaded me to seek a camera with a bigger sensor that could capture the image out of the photo tube – projected directly on to the sensor with no intermediate optics. It is hard to beat air for not degrading an image, and purely mechanical adapters can be quite cheap or 3-D printed. The photo tube and adapter, however, have to project the intermediate image somewhat less than 20mm above the top flange, or be made short enough that they do.
A few years ago, a used micro four thirds camera was the cheapest source of a suitably sized, high quality sensor, in Germany for my microscope (ca. 110-150 euros for a camera with fewer than 100 shutter actuations and often a one year guarantee). These days, if I were buying in the U.S. – where the OP is? – I might consider, say, the Nikon 1 J3 to see if it would answer. It has a one-inch sensor, an electronic shutter, live video out, a shorter flange focal distance, and as little as a $100 sell price on ebay.com.
Anyway, just an idea that may not work. I really do not know Nikon cameras, the OP's microscope or the optics of its photo tube.