Thanks for the inputs!
Unfortunately the pictures and the gerber viewer don't serve the purpose I had originally in mind. This being a completely unknown PCB to me, I wanted to search for the component name to find it quickly. Not very important though.
Looking at IanJs video I note a few things:
* he has a similar issue, although at lower currents.
* Recovery goes very fast until it almost reaches the set voltage, then it takes considerably more time. Is that normal...?
At least that's something I can't see on mine. But I can't go that low in current without hitting the heartbeat issue either...Edit: Sure, same behavior on mine, too. Still: is that normal?
* We're now at at least 3 DCP405 with pretty much the same issue

@IanJ: how about your other DCP405?
Progress on my side!
As far as I can see I have no missing component on the entire board when comparing it to the PCB images on GitHub. For quickly checking component orientations it would be nice to have a high resolution picture without heatsink.
Disabling the down-programmer "solves" the issue! I attached two screenshots that show the output voltage after enabling the output, one with down-programmer enabled, one with it disabled.
Kleinstein suggested to change the current reading to the lower range, too. I don't know how to do that? I have set the DCP405 explicitly to the 50mA current range but I'm still getting only 0.5mA resolution on the read out value - even if my set value is 10uA.
Kind regards,
Max