best budget hi-fi speakers for my specific system

I doubt that you will be able to find Vandys 2c for the desirable price used. Becides, these speakers need heluva amp to drive them. You can get Vandersteen 1c used for that price(quite reasonable)
I used to own 1c and they're perfect for budget high end or beginning 2ch. Dare to say that they also love high power but not as 2c.
The other suggestions about monitors sound also good for your idea.

Soliloquy 5.3 on sale on A'gon for $650 will get you the detail and warmth you're looking for.

I would recommend the Totem Hawk Speaker. The Hawk speaker images well, produces a large soundstage, is very transparent and musical, and detailed. Very difficult to find another speaker in its price range. The Hawk speaker is affordable and plays much bigger than its price and size. Lastly, the speaker goes well with solid state or tube gear.

I have owned Vandersteen 2c and 2ce's for many years. I also currently own an Audio Refinement Complete ("ARC") but it would not be my choice to drive the Vandersteens 2c's with it. I mainly preferred tube amps or hybrid amps with the Vandersteens. I downsized my system last year and the ARC amp was part of that move. I originally tried the ARC with a small pair of Monitor 4 speakers that I had been using with an NAD receiver and the NAD was a better match. I had the same problems you describe with the midrange. This led me to sort of a temporary fix (until I have more time to audition more stuff), B&W DM 602S2, which work ok with the ARC for now. Once broken in, the B&W's are a richer sounding speaker than the above mentioned Monitor's. You might be able to drive a Vandersteen 1 with the ARC as the Model 1 is a more forgiving speaker than the 2c (and a used pair should be in your price range). Many people (myself included) read the reviews of the ARC which claimed it to have the qualities of tube amps. The ARC has a lot of good things going for it but it does not sound like the tube amps that I have used (Quicksilver, Berning, Moscode).

I would recommend giving the speakers @ least 300 hours of playing time before drawing any conclusions as to their sonic signature (the little $300 AE's take this amount of time (plus) to run in and afterward they evolve into great sounding budget speaker). The speakers should also play much louder, without straining (or closing in) after this period of time. In the meantime, do not push them too hard. If you do not care for the sound @ this point, then look elsewhere, but @ least give them a chance. It would also seem to me that a metal dome tweeter might be a decent match with and AR Complete integrated amp and CD player (if these are the components which you own) as the conbo is not at all "hot" in the HF's (I have owned the player myself and have listened to the amp).