First thing, Go to the downloaded sketchup installer .exe (if you don’t have it download it again) select it, right click on it and choose Run as administrator. Follow the prompts and choose Repair when given the option. Once done, reboot your PC and check if spacemouse is working in SU.
If not,
Go to C:\ProgramData\SketchUp\SketchUp 2022\SketchUp\Plugins
select the folder and file both named 3DxSketchup and cut to clipboard,
Go to C:\Users\Your User Name\AppData\Roaming\SketchUp\SketchUp 2022\SketchUp\Plugins
and paste the folder and file in there.
Open Sketchup and check spacemouse.
As a side note, when filling in your profile Windows isn’t enough, I have guessed you are on Win 10 but these instructions would be different for win 7 or win 11.