So let's recap, you had that problem with the AV7005, now with the Integra, yet no problem with Anthem prepro? I have used quite a few pre+amp, prepro+amp, AVR+amp combinations in the past several years using Denon, Yamaha and now Marantz products without any hum or noise issues. I really don't think it is a particular brand related issue. You mentioned before other AV7005 owners experienced similar issues but that seemed quite normal considering the fact that the AV7005 is likely one of the best sellers in recent prepro history. I google searched hard after what you said and found a few, just a few that reported such issue, that is not out of the ordinary.
From what you are telling us now, it would seem to be a different issue than ground loop because it only hums when trying to do DD, but then what is the odd that your loaner is doing the same? This is beginning to sound like a mystery. If and when you finally find the root cause please do share the information. I am sure I am not the only one who really would like to know what the heck the problem is. As far as connecting the Anthem and Integra chassis together, I doubt it would make any difference if the problem only shows up in DD, but you can try. As you said the Anthem has a screw for such connection but it may be hard to do with the Integra as you sure don't want to scratch off the paint job.
Have you tried the suggested way to connect one thing at a time to try and narrow down which piece of equipment would bring about that hum? Again, as ADTG suggested, and you have already tried that, i.e. the cheater plug, so ground loop really is not likely to be the reason.