The renaming I was referring to meant the name of the player and subsequently the name of the Pi. Since the preinstalled instance of Squeezelite takes its name from the Pi’s name, you should just change the name „max2play“ that is displayed in the menu „Settings/Reboot“ of the M2P web interface. After saving and a reboot, the name of the device and of its Squeezelite player should have changed in the LMS interface.
As mentioned in your 3., this should work. I just confirmed on a test device to make sure. Please try following the steps mentioned above.