The microscope already streams the camera feed to any remote client on tbe local network. Even locally on the Pi the microscope interface is using http requests on a virtual network. That client could surely then forward it to youtube or anywhere?
There was a discussion here somewhere. I’ll try to find it.
Edit: It turns out the other thread is from @kenzoat, so the same application. Accessing the Raspberry Pi Camera Openflexure Connect video stream. And my response was the same as now 
. On that thread @r.w.bowman had some more specific comments on implementation, but clearly you have not found that to work for you?