Someone else may have a different opinion, but my recommendation would be for you to get a used DSLR or mirrorless camera and NDPL(1)-2x microscope adapter with the appropriate mount for whatever camera you get. I’d recommend that solution for the following reasons:
Unlike a dedicated microscope camera, a DSLR or mirrorless camera can double as a high-quality camera for general photography (if that is of any interest to you)
My (admittedly very limited) experience with dedicated microscope cameras has been full of frustration, while putting my DSLR on my microscope has mostly been very easy (but YMMV)
The right combination of DSLR or mirrorless camera and software will give adequate quality for streaming video, and will give better image quality than a dedicated microscope camera if you ever decide you want to get more serious about video or photography through the microscope
I’m not sure where you’re located, but in the US, one can pick up a nice used Canon camera (one of the better brands for microscopy) with video capability for ~$200, and used NDPL(1)-2x adapters sell for ~$50–60 on eBay (and new ones are $90–120). That easily puts this solution within your budget. These type of articles may also help.