Hi there.
You might already know this, but one workaround for having different behaviours is to duplicate the relevant page and amend it to look and work in the way that you want rergarding the menu.
The duplicate can then be set to do not display on all breakpoints except the desktop one. The original can be set to not display on the desktop one only.
This way, you effectively quarantine the amended page so that it does not mess things up on other breakpoints.