The UR22C may not have the greatest DAC there is, but honestly it should be
just fine here (it's getting about as much dynamic range out of a CS4270 as you can reasonably expect, ~106 dB(A), and there is an analog monitor gain control). I would look into software EQing the hell out of these HS5s, 'cause
they sure do need it. Low shelf of +3 dB / 450 Hz (or -3 dB high shelf at 450 Hz), -3 dB at ~1 kHz at Q ~= 2 or so, +3 dB at 2 kHz with Q = 1, pre-gain -3 dB to keep levels in check, plus -2 dB high trim on the speakers themselves, and go from there. You can potentially do this completely for free (e.g. Equalizer APO + PEACE on Windows if you can live with shared mode playback), things get a bit more spendy if you need it in a DAW with e.g. Sonarworks (which comes with a decent measurement mic though, equally useful in REW and can be used for attacking room modes either way).
The ADI-2 DAC will also do parametric EQ in hardware, but before splurging on such an expensive piece of kit on the DAC side I'd much rather be upgrading the speakers instead, which are quite clearly the weakest link here. A pair of Genelec 8030Cs would be roughly the same price here, and I'd much rather have those on a UR22C than the HS5s on an ADI-2 DAC FS.
