Logitech G502. You are right, I'll try it later today.
I have that mouse as well. I don't have any special software for it installed, as it seems to work nicely without.
Given that we both have the mouse, maybe we can figure out what is going on.
So, I found out that the issue is full screen. I don't know why, but my mouse doesn't like Dosbox full screen. Try Alt Tabing out of full screen and you probably notice that mouse suddenly works fine.
So I went to edit dosbox_warcraft.conf file and changed "fullscreen=false" and "windowresolution=1920x1080" or whatever is your desktops resolution.
Still can't play full screen, but big window is doable.