LinearMouse already has built-in actions for this. In the button mapping actions there are Mission Control actions named Space Left and Space Right, and the action handler posts the corresponding macOS symbolic hotkeys.

Map M4/M5 under the Buttons settings to:

Mission Control -> Space Left
Mission Control -> Space Right

If those actions do not fire in your setup, use Keyboard -> Key Press instead and record the macOS shortcuts for moving left/right between spaces, usually Control+Left and Control+Right. Also check that LinearMouse still has Accessibility permission and that those Mission Control shortcuts are enabled in macOS Keyboard Shortcuts.