I have owned various E39 Chassis vehicles throughout the last 6 years. I have experienced this on three of them. I beleive it is a tweaked door handle itself, that causes this issue. There are two ridges on the moving handle piece that rub on the stationary metal piece that mounts in the door. I have successfully filed down the stationary piece that is interfering with the door handle itself. They are aluminum handles, and I beleive they are softer than one thinks.
It is easiest to fix this by removing the handle from the car. This involves pulling the interior door panel, foam weather vapor seal, rolling window up all the way and sliding the detent tab towards the fron of the door by about 3/4". It can be done through the access plug bu the door latch with a special tool from BMW. I have his tool, and it is still easier to remove the door panel and get at it straight on.
You could also try looking under the handle and see where it is scraping, while actuating it. Try tweaking the handle the other way by a little bit. It will not take much at all.
Lube will not help, as the squeak does not come from the pivot itself.