Windows 10: Use a second mouse just for scrolling?

As for the question: I think the answer is no. If you want to emulate system-wide scrolling (mouse wheel events) using the second mouse - I think it can be done only by the driver (or some driver-like software) - so if there is no such driver/software with such options for your trackball or mouse, then you're out of luck. Though it is quite plausible that such software can be written or even exists already (I don't know).

As for the general problematic of scrolling, and since you've mentioned RSI: I feel sympathetic for this problem. Yes it is quite stressing to scroll with the mouse wheel and this is needed very often - browsers, editors, etc. So this is a real problem.

Fortunately there is an excellent and free solution for this. There is an app called Autohotkey and it can emulate scrolling in a way similar to what you describe. I have made a script which emulates dynamic scrolling with mouse movements. I've been using it for 2 years and it's stable and well tested, and it's waaay better then the wheel! I just can't live without it now.

Here is github link with thorough description and how-to:
https://github.com/Mikhail22/Autohotkey---Mouse-scroll

It does not answer your original question, but can solve the scrolling problem. Hope this helps.