bookshelf for classical music under $2000

I agree with Mateored: Avoid metal drivers, especially metal tweeters. Look for soft-domes (plastic, cloth, etc.), ribbon tweeters or the revived folded ribbon (a.k.a. Heil air motion) tweeters. Also, the sound that bothers you, which also bothers me, can result from poor crossover design or implementation, so just having non-metal drivers is no guaranty of a winner. Some brands to audition:

Reference 3a
GoldenEar (although they really don't have a "monitor" yet)
Silverline Audio (good luck finding a dealer)
Dynaudio
Mark & Daniel
Quad
Kirksaeter
Morel
Sonus Faber
LSA
Nola (Boxers)
Vienna Acoustics
Fritz
Spendor
Opera
Anthony Gallo Acoustics

Great list. I might also add Merlin TSM in one of its various iterations to the list. They come up for sale fairly regularly, used, these days on this website. They are very revealing and resolving and have a fabric dome Tweeter and are definitely not harsh or overly bright. They were, ultimately, not my "forever" speaker, but they are worth listening to. Worth talking to Bobby at Merlin to see what he can do for you. As for Reference 3a, I currently have the Dulcet which, brand-new, is within your price range and very special, indeed. In either case, check out some of the reviews online. Hope this helps.

Try Ushers

Can I ask why only bookshelves? Compact floorstanders could give you more bass per dollar, with no need for stands.

Also, have you got any acoustic treatment in your room? And have you experimented with speaker placement. Brightness can come from side reflections and excessive toe-in.

I agree in principle with several of the brands posted above: Spendor, Dynaudio, Nola.

This widespread perception about metal tweeters is crap and has taken on the status of audiophile myth along with direct drive turntables and MM cartridges. Metal dome smoothness depends on how the dome is damped among other things. I have a low tolerance to harsh treble, but my Mirages have a treble that's sweet as silk--powered by titanium domes. They use a cloth suspension to damp the tweeter and it works. Because of the light diaphragm and titanium's resistance to breakup, the titanium tweeter is both articulate and very smooth with no artificial high treble boost or edginess.

That said, there are lots of great fabric tweeters as well. It seems the Nola Boxer would be a great choice for the OP's criteria. Also, GoldenEar DOES make a pair of monitors, the AON 2 and 3, which are stand-mounted and use a most excellent Heil-inspired folded ribbon tweeter.