Shove a strong magnet up a PEX pipe and glue in place. Use it to probe with it. I have one made for working on my car, like when I drop screws and tools in places I can't reach. It works really good.
Something I'm curious about though... if you blare music into the air handler, I wonder if you can hear it at the covered registers
Either ways, if I was in that situation, I don't want to get up and do any work. I'd pick up my camera and phone. Document. Make 'em fix it. If they don't, there are paper means of getting it fixed on the pocketbook of whoever screwed up unless you used some unlicensed monkey in which case you have to go pound sand.