Thanks for the links! There were two that I hadn't discovered in searching the forums before posting.
There was ≈200mV P-P visible noise to the VIN ground on the +5 and the + 3.3V, clearly originating in the Vin regulator. As shipped, no amount of added decoupling to any available ground would remove it. The +3.3 is hard jumpered as AVREF, so not much to do there without brain surgery on the jumpers, which I'm not well-equipped to do. I'm also not sure it would have helped.
The MPM3620 regulator on the DUE I had is spec'd to operate at 2MHz, but when the light load startup funk was going on, I could hear something squealing at a high, but audible rate. There's no physical inductor that could be expected to be microphonic, so the source had to be one of the two Hi-K chip capacitors next to that part. There must be some large ground loop currents flowing around that regulator.
As a foray, I ordered two "OSOYOO DUE R3" clones from Amazon for $21 each. Those came in today and both show noise just over an LSB in 12 bit resolution. The clones use the LM2734 regulator and 100 uH discrete inductor of the original reference design. There's 100mV ripple on the +3.3, but its a 500 KHz waveform rather than a collection of noisy pulses.
So I have a fix, sorta', - at least something that works without degrading the dynamic range of the project, and at a low price. Thanks for your advice.