Both computers are running Windows 10 and are fully updated. The primary computer is running a two monitor set up (1920 x 1080 on the right, 1680 x 1050 on the left), and the secondary monitor (seamless to the left) is running 1920 x 1080. Now that you've mentioned that, I've tried switching the seamless control boundary; if the secondary computer is accessed from the primary computer's first monitor (same resolutions), no clicking errors happen, but if the secondary computer is accessed from the primary computer's second monitor (different resolutions) clicking errors happen.
This does fix the problem, mostly. Thank you very much! If there's any way to make that not happen without changing the boundary location, that would be good to know, but until then I can at least control the computers without the glitch.