rouge-impregnated Fabulustre cloths that I (rarely) use to put a higher shine on a bezel or buckle (e.g., before photographing a used/scratched watch). They're available for a few dollars from watch supply stores.
The only caveats I'd have re cleaning are (1) don't use anything more chemical or abrasive than we've already discussed, and (2) don't have anyone other than a Lange watchmaker "professionally" polish it. If they try to polish it without removing the crystals, you might lose parts of the anti-reflective coating, which won't look good and will necessitate crystal replacement (at a cost of about $200 USD/crystal). As I posted on another Lange forum, below are pics of what this looks like (and also what the movement looks like when a non-Lange watchmaker tries to clean or open the caseback -- you can see small blemishes and dust/hair). A European dealer, who said this work was done by "a watchmaker who used to work for Lange," commissioned this unfortunate bit of "watchmaking" just before he put it up for sale.