The
Shure circuit worked! See circuit diagram in previous post, or that link. I had a scrap piece of pcb and some 47 mfd caps, so I soldered it up... quick, dirty, and rather ugly.
The noise floor is now the same as with the battery. I plugged the guitar in (with internal mic) and it sounded like a mic'd guitar!
It will take some time for me to get a better breadboard and enclosure design, etc. But it works!