Anyway since you got both desoldering station and Atlas tester, remove suspects to test. You will find what is wrong. After fixed, make sure that the drop across R1 confirms your chosen CCS setting.
I suspect strongly is the BIII because Brian said changing the DIP switches doesn't affect the voltage but with one configuration of switches Reflektor-D worked like a charm only isue BIII outputing DC with DSD material.
I am clueless about BIII and its switches but maybe your Ref-D is OK too and the sense wasn't connecting? Convert it in two wire mode to make your tomorrow tests simpler also.
I changed RX 150R in place of 100R.
Dummy load
12R Vout after 1/2 hour warm up 5,355 VDC
8R Vout after 1/2 hour warm up 5,309 VDC
Dummy load
12R Vout after 1/2 hour warm up 5,355 VDC
8R Vout after 1/2 hour warm up 5,309 VDC
Ok now we know your DUT and its settings are good for 660 mA. We could go on loading it further if you have the sinks for it or we could feed the BIII with another source and see what it consumes.
Russ told me yesterday to take out the controller, all DIP switches in position ON. I only have S/PDIF 4 + Sidecar.