I use the Bearsford DAC and I have the exact same problem. If I have the volume low I can hear a very high pitched whine as well which drives me nuts. In my case it's not just isolated to Spotify, it is the same if I use ripped CDs through iTunes on my MacBook Pro. This points to a problem with the DAC I think . Very frustrating.
Source: Heavily modified DDDAC using MacBook Pro.
Amplification: Pre amp-Stereo Coffee pre amp on steroids.
Power amp- Rather large Colin Wonfor Class A power Amp
Speakers: Lampizator P17 Open Baffles
Accessories: Klotz 500 DIY Interconnects & DIY speaker cable sounding better than anything commercial.
All powered from a Mother Trucker BPS with a dedicated consumer unit with 10mm armoured radial back to the house tails after the meter.