Dear Andy,

Thanks for the interesting problem.

>   The resulting radiation pattern should have the nulls filled
in, and be a reasonable approximation to omnidirectional in azimuth.

Would it? I'd expect there to be directions in which the fields from the
two loops were opposite and equal, giving nulls but in different
directions.
Specifically, I think there will be two directions at 45 degrees to the
planes of the loops where at any instant you get (to put it crudely) a
north pole from one loop, at 1/sqrt(2) of its max strength, and a south
pole from the other, also at 1/sqrt(2) of max strength.

I don't know about efficiency, though - you might have started an
interesting technical discussion!

73,
Chris G4OKW