So how are you handling the balanced/unbalanced issue?

These designs are in the ARRL Antenna book. Generally a second loop is
used to couple to the first loop to act as a balun. [Mutual inductance
between the loops couples the signal.]

The shield should be broken at the far end of the antenna, i.e. not the
end that goes near the radio.

Since you are nulling signals, I gather the loop is vertical, but 40ft
in diameter seems pretty hard to believe. We are talking 3 to 4 stories
high.