I don’t have this mouse but check the following things as I noticed one of my mice doesn’t have exposed screws: 1) case opens from midline assembly (gently pinch top or bottom half and see if it can pull apart 2) top of case has a thin top which buttons are set into and can pop up and out to reveal compartment 3) recessed screws are hidden under rubber bottom “pads”. Peel up pads to locate screws.
I just ran into the same thing with an iPhone with a cracked screen and I couldn’t get the suction cup to stick. Instead of a plastic pick, try a feeler gauge or rounded blade to get under the screen. It’s a PIA but you can do it.
Check the relays that are ignition / starting related and fuses nearby. Clicking makes me think a relay coil is pulling in but perhaps the contacts aren’t plugged down (relay is not all the way engaged (pressed) into its socket) or an associated fuse is not pressed down. Basically pull them out again and carefully reseat each one back into its socket.
Take a look for a lighting relay or fuse especially if it doesn’t matter which door is not triggering the lights. Most likely the wiring from each door switch comes together at a relay and the relay contacts are what enable the lights to come on. Sorry I don’t have details on the specific model but you might be able to locate one or two fuse/relay boxes (normally one under hood and possibly one below the steering console around the gas or brake pedal area. Good luck!