Hi Ross: I think the AIR pump is one of those few but common problems that the 500E/E500 has, such as leaking power steering boxes. Like yours and others, mine makes an annoyingly loud clicking noise for the first 5 minutes of operation until the clutch disengages
It's a pollution-control device but not actually needed for reliable engine operation, so my mechanic pulled the plug and disconnected the darn thing. Noise went away but after several thousand miles the "check engine" light came on. Re-setting that is a minor annoyance because a special tool is needed and the battery must be disconnected. After going through that a couple of times I am now willing to put up with the tick tick tick noise.
Up here in Canada, you are only required to maintain pollution-control equipment for four years, which coincidentally is the length of Mercedes-Benz's warranty on the item.
Let us know if you find a cure that doesn't involve buying a new pump.