Metrum makes their own DAC chipsThey may make their own discrete R2R resistor boards, like MSB Holo Denafrips ect, but I doubt they have their own R2R chip IC’s manufacturing processes.
I think you are on the right track with R2R. As others say a lot depends on the other components and source materials but I find they are more "analog sounding". I had chip-based dacs, went with at TT and LPs to get that analog sound, then switched to an R2R day and wish I'd done that first as I wouldn't have gone with analog sources. Sounds amazing. Tough to fit in your budget though but have a look at Kitsune' / Holo Audio. Maybe the low-end Spring 2...
I guess its too late but if you have a chance to listen to an Audio Note Dac 0.1x it's in your price range and might be just what you're looking for.
Don’t expect too much from a DAC change. I’ve had a Chord Qutest for over two years, while my system has been through many upgrades, and changes to position and the room. At the beginning, I could never take the Neutral Incisive filter as it sounded “cold and clinical”, and I’d move to the Warm, which introduces distortion and makes the sound pleasant, but mushy. It sounded better until recently, when the rest of my system got right. Now the Neutral Incisive filter sounds musical, clear, deep, tall, and with realistic timbre and that “wow, the instrument is there” feeling. I’m not trying to say Chord is the best, but I want to warn against the endless search for “warm and lifelike” in a piece of equipment, when the system as a whole makes that happen.