Same happens with my 5F4 copy, and less-so with my 5E3 copy. And only does it with my Fulltone pedals; as you say, non-true-bypass doesn't make the click.
And it doesn't always do it either.
Take a look at
this. R.G. Keen now designs pedals for Visual Sound. I can take a picture of the old labels on my Fulltone stuff (from ~2000), but I noticed they now claim a "proprietary anti-pop circuit" in their pedals, meaning they took steps to fix the problem in the pedals. Alternately, Jack Orman suggests it might be a problem with the LED power circuit, and
offers a fix.
So why is it more obvious with your Super than with his other amps? Most likely, if his other amps have gain channels (and he uses them), it's because the Super has a wider dynamic range because the output tubes will be the first thing to distort. Because of that, the pops do not get squashed the way they would in a distortion channel as the rest of the guitar signal is being clipped.
Once I played my 5F4 copy alongside a guy with a 50w half-stack. His amp was pretty well cranked, and I had a Full-Drive and my amp turned up to ~4-1/2. I sounded like I was blowing him out of the room. The reason was my sound wasn't getting compressed (amp wasn't loud enough for output tube distortion, and the pedal was set in "Comp Cut" mode). By comparison, he was playing with a preamp gain control set very high and the master down, and had a lot of distortion, but also a lot of compression and sounded muted.
He could've stomped, clicked popped, etc and you'd barely notice it through that setup, but a clean and dynamic amp would make it obvious.
I still say the problem is your buddy's/customer's pedal. Have him bring it along and demonstrate the problem. 85% chance he won't replicate it in person with you. If he does, look to see if Keen's suggestions were already followed in the pedal. If not, mod the pedal. Stay towards the high end with your added resistors (4.7M @ 1/8w-1/4w sounds good).
