Regarding whether the bike shop should help you out, I think they should. However it's the sort of thing that makes people wield their own tools as the bike shop doesn't ride the bike and will never be as focussed as you are on getting it right.

It's not the sort of problem that sounds as if it has an obvious answer. As you've suggested, the answer is to work through the possibles and eliminate them one by one.

I'd suggest starting at the front deraileur / chainset / chain and if possible eliminating these from the enquiry. Then it boils down to the deraileur chain of command or the cassette not working. Key is to check everything very thoroughly as it's probably something quite basic but subtle, like the time my riding buddy's rear deraileur shifted into a bigger cogs as the ride went on. Turned out the downtube and bottom bracket lug had separated making the path for the cable longer. It took us at least 15 minutes to spot it as we were focussed on the deraileur.

Some ideas off the top of my head, which may help.

Pre ride checks;
Is the chain roughly the right length?
Is the deraileur itself and hanger straight, and hangs vertically below the cogs when you look at it from behind, with the upper pulley in the middle of its float?
Are the deraileur pulleys set up the right way round, i.e. floating one at top?
Are both deraileurs set up with correct endpoints and b-tension?
Is the chainset / bb combination correct to give the right chainline? Are the chainrings set up with correct spacers (if needed)?
Is the chain lubed sufficiently and no bent links, 8spd quicklink or other craziness etc?
Is the deraileur cable friction free?
Measure all the spacers in the cassette - could be a 10 speeder in there, clean and reassemble. Is lockring tight enough?
Are the STI levers tight on the bars? Are all the cable end points on the frame firm?

When you ride it you could help diagnose it by:
does pulling the deraileur cable / adjusting the adjuster make the noise go away?
does it make the same noise in the middle ring, or just in the big ring?
does it make the noise just riding along, when you press the pedals, when you get out of the saddle?

Desperate measures;
Swap out each part in the drivetrain to see if it makes any difference
Measure the deraileur movements using a dial indicator and if different to the cassette spacing get your machine shop to make up a custom spacer set.

You're not going to solve it;
The tandem has really short chainstays and therefore with a triple there will always be a degree of compromise about the shifting?
The tandem flexes a lot when you go for it (both unlikely I think).