A warm DAC?

My highest recommendation for your budget goes to the Lampizator Atlantic. I suggest you contact Lukasz Fikus directly @ lampizatorpoland@gmail.com.

Musical Paradise MP D2 mk3.  User can roll both tubes and caps.  And you can use both 12 or 6 volt signal tubes and 5 or 6.3 volt rectifier tubes.  Capacitors are not soldered, just get installed with thumbscrews.  Options include TOTL Crystek clocks and interchangeable AKM or ESS Pro chip sets.  You can find your way to any type of warmth or "musicality" you like.

You may want to check out the Sonnet Morpheus R2R DAC. It’s not on everyone’s list but it’s a wonderful DAC and could be considered “warm” when compared to other R2Rs...

@larseand ,Head over to Audiocircle and check the threads in the Discless Circle for the Musical Paradise MP-D2 DAC thread for owner reviews.

I was in the same search as you a couple years ago and took a chance on the MP-D2 MkII.
You can spec the DAC chip (I'd recommend the AK4497), and run a full range of tubes options. I've settled on a NOS Westinghouse 5U4G rectifier and a matched set of NOS Amperex JAN 7308 tubes. The sound is glorious and towards the full, warm side of the spectrum. Swapping driver tubes to a set of NOS BEL ECC88, yields a very different signature (lighter, more sparkle up top). So, the tube compliment plays a big role. All in at about $1500. You could spend the rest on nice cables. It's worth looking into this hidden gem.

bel canto 2.7 or better yet the 3.7 are silky smooth / musical and all the micro detaisl and dynamics without the digital hash, rigging or sterility.