Depends on your music preferences of course. I had a very similar system to yours: BAT VK 60 monos, Cary SLP 2002 and a BAT VK 5i and Watt Puppy 5 speakers. My system has changed from this, but you get the idea.
If you like smaller scale recordings (small jazz groups, accoustical rock/pop, female vocals, etc. . .) the Audio Aero Prima is a great player for this configuration. This player does a fine job of larger scale recordings but the sound stage looses its focus a little bit with larger recordings. I owned this player and sold it recently for a new Resolution Audio Opus 21 with GNSC mods.
Also, based on the rest of your system, you could also consider a Wadia (like the 830 for example) which I also owned. This unit is certainly more detailed than the Audio Aero, but the rest of your system is warm enough so that it would be a good match. Wadia's are very well made and very reliable (nice for living in Alaska).
Both of these are in your budget. If you wanted to spend a bit more the Resolution Audio is very nice sounding, especially with the mods.
My experience with the Cary tubed CDPs is that you may end up loosing too much resolution within this system of yours and may not ultimately please you. IMHO. I think this may also be partially true with the Lector and the Bluenote from what I understand* (*not having listened personally to either in my system).
The AR players, especially when you get up to the 3 are very good. I think you ultimately may want to go solid state though as your system has a fair amount of warmth already between the Cary pre and the slightly less warm BATs. There is no way your speakers should be sounding bright with this system.