Did you leave it overnight so that it would start up cold?? Or was the engine warm when they diagnosed it?

If it was the tensioner shock, it would make the noise all the time, not just when cold. The air pump only runs for the first 30-45 seconds on cold startup until the cat converter is sufficiently warm to do its job.

I had the same EXACT symptoms as you on my 300E and it turned out to be the air (smog) pump bearing gone bad. I simply disconnected it.

Good luck!

~Paul