Rogue Audio M-120 Magnums...first rate in every way, I own a pair.
Another possibility is to keep the 4BSST and replace the preamp with a good tube preamp like a Rogue 99 Magnum. With it you can tube roll 'til you get the sound you like. I had a 99 with my 4BST and it was warm and rich yet detailed and very dynamic with the punch of the 4B; excellent detail and articulation at the extremes with a rich, smooth midrange. I finally upgraded yet again from the 99 to a McIntosh C2200 tube preamp and paired with the 4BST and the combination almost ideal. You can probably get a used 99 Magnum for about $1500.
I'd do a tubed preamp first..to test the waters, instead of putting tubes next to a Bryston pre...
I second Dave's suggestion on the Rogues, bang for your buck/entry into tubes, and cheap! Or eventually may want to look at a VTL ST85, older mb120's, or newer mb125's..
p.s. I used to have a 4BST and 3BST with a BP5 myself...blow your mind a go tubes..
Aronov Audio tube integrated LS-960I is way below your budget as used item, and maybe even when new. You will still hear the best bass slam and clarity emulating solid state, but with pleasing musicality throughout the full frequency range, and a very natural or realistic sound, minus the bright edge. (Did this win the longest sentence contest?)
I'm fairly new to this hobby but I would suggest trying a good quality tube pre-amp that would mate well with your bryston. I tried this and found that it took some glare and brightness off without diminishing the controlled bass that I was used to. You could also try a tubed cdp like the Jolida and see if that takes away some of the digital brightness (and still have a grand left to experiment with some tubes and upgrade the cord on it).
Just some thoughts, as I've moved into tubes recently.
If you like the sound of the Bryston amps, then I would stick with them. I like their amps for what they do but never liked the preamp. So yes if it's replacing one or the other, I will definitely replace the preamp. You might find that even a better SS preamp would give you the sound that you need. But audition the tube preamps mentioned and see.