Oh, I bought one second hand a few weeks ago and forgot about it till now!
It's a 'Bigshot ABY', a little different to yours.
Mine isn't battery powered (for lights only), is yours? Have you connected a power supply?
does the clicking only when you toggle between amps? if youve got no lights and it clicks/pops when you hit the switch, swap out your power supply or change the battery. it sounds like the buffer doesnt have sufficient power. If you don't have power, the switches will still work because they're mechanical, but the lights and buffer won't function.
Does it still click regardless of what polarity it's set to or whether ground lift is on? I don't think out of phase amps would make that noise, but you never know. The lights not lighting are indicative of some kind of power issue, like maybe if the power supply is plugged into the same power source as the amps - could the clicking be a 50hz AC issue? Have you tried it with a 9v battery? are both of your amps properly grounded? What's your setup look like - are you running other pedals or into a powered pedal board?
Sorry for the lack of answers, just wondering out loud what variables in your rig could be contributing to the issue (if any).
I have the bigshot, which is the passive version of the twin city with no buffer. Ive never had any problems with it.