The "600 Series" B&Ws are a value-driven, high bang-for-the buck speakers that compete very strongly in their price class. What you are experiencing is not so much a "tweeter" issue, but the aggregate of good, but not great, crossover components, signal path, etc. The CM series is a major jump in sound quality with point-to-point wiring, better components, and cabinet integrity. The 800 series takes this up a notch with a great deal of attention on refraction(s) and parts quality upgrades throughout.
At your price range, new CM series would get you there, or used 800 series. I’ve also auditioned Totems in this price range and found them to be very musically satisfying.
Good luck in your search. Have fun.