You may wish to check what the lenovo thing checks -- if it is an OEM machine thing then you might be able to fake that at some level for its purposes. I doubt they would have gone hardcore DRM or hardware unique concept for this or be checking BIOS signatures like Windows activation (by the way activation tools might be a) needed and b) make for an easy way to install OEM signature stuff) and instead might only check for a logo or certain registry keys* being present.
I don't know that I have seen it done for a mouse utility before (normally most people want them uninstalled or sidelined) and instead it is usually more for graphics (granted that is usually more make generic drivers work when the OEM can't be arsed to update them any more), wifi utilities or audio managers.
*there are tools that will monitor where in the registry is being read without having to go hardcore hacking mode if you have a working thing available to install and uninstall on, or maybe copy from working install to new machine and replicate functionality.
https://windowsreport.com/monitor-registry-changes-windows/ looks like a reasonable start. Your anti virus if you crank it to paranoid mode might even want you to OK every change which works as well.
I can't say I have heard of such a program before, mainly as most things I care about here are more for games (which tends not to make it further than mouse jail and macros), but I would check around gaming forums, accessibility forums (I could see something like that be useful for those that might not do precise motion), multi screen forums (while things are better than they were I still get things want to split the difference).
That or maybe just knuckle down, bust out a copy of autoit, autohotkey, whatever gamers are using for macros (could be a general language like python or lua, could be something more specialist). If it is a more macro/automation focused language (or library within it) then moving a window is probably a code example. Moving it to position ? where position ? is screen resolution divided by 2 (or indeed might have some kind of middle of screen functionality baked in) whilst leaving or restoring the mouse afterwards is then a minor tweak on top of that. Or if it is a window that will/you can make spawn at a certain size and location then a snap to and move macro is possibly not even coding.
