Audiojan,
I went through this same exercise back a the beginning of this year, and listened to some great preamps. You're wanting to keep the cost down to about $5,000 will knock out a couple of great options: BAT VK-80, and the Backert Labs Rhumba Extreme (not the base Rhumba....it won't compete). In the end I chose the Aric Audio Motherlode XL as I wanted the XLR connections, but if I didn't then I would have ordered the non XL version. Several months after my purchase, I am still extremely happy with the Aric Audio preamp and would do it again.
If you're looking for a "little warmth", I'd suggest a 6SN7 based preamp personally. I'd look at the already mentioned: Aric Audio, Supratek, and the Don Sachas/Lynn Olson Raven preamp. I'd highly suggest you reach out to Aric Kimball (Aric Audio), Don Sachs (Raven preamp, sold by Spatial Audio), and Mick Maloney (Supratek) to have a conversation. Aric Audio and Supratek make a bunch of preamps, they could lead you in the right direction on their products.
For transparency, I own an Aric Audio Motherlode XL preamp.....and Aric certainly can customize his preamps, and he has adjustable gain controls.