If I can make a few suggestions here....
If you have another computer available, can you plug the mouse into that and make sure that the buttons are functioning normally on a second computer?
Can you uninstall the HP drivers completely, reboot your computer and while rebooting, unplug the mouse. My recommendation would be to remove the software, and then open Device Manager, and then remove the device completely from the "Mice and other pointing devices" listing (see attached screenshot). Be sure to check the box to delete the driver. Then reboot.