These kind of issues are difficult to solve remotely and often need hands on experience.
I doubt that clocking would be an issue as it cannot be changed in the DAC AFAIK in any case so if it is the issue there isn’t anything that you can do. However in your case the clock will be derived from the incoming S/Pdif data.
Thinking back to the last century ( !) I do recall having a DAC that would inject a short burst of digital noise every so often and that resembles your issue. The solution was to change the digital interconnect, in that case between DAC and transport. The shielding of the interconnect seems was not to have been working as it should and I think that the cable/DAC was picking up switching from a neighbouring apartment ( fridge?). I presume that the CD transport uses another cable hence no problem.
So you could just try another cable - no guarantees though.