@joe_kohli Short answer, no, your mouse button software does not know how to activate an AutoCAD command, and CUI command does not have hooks into your mouse driver to override the mouse driver.
But... your mouse driver has many keyboard button functions that can be assigned to the buttons you want to customize: find those key combos, set each button to a different one, then use CUI command to assign MOVE to one of those key combos (assuming it is not doing anything else).
Just remember, in ALL programs on your PC, an in windows, those buttons are going to activate that keyboard key combo all the time. So choose wisely.
Autodesk offers this very abbreviated post on the topic
https://www.autodesk.com/support/technical/article/caas/sfdcarticles/sfdcarticles/Assigning-AutoCAD-....
And don't be fooled by this find in CUI command, it's a legacy leftover for those still using old-school tablets from decades ago, it's non-functioning for modern mouse/drivers.

HTH