Since the inductors looked ok in Fitting L and R values, I decided to check the capacitors and the LC tank circuit using the same technique. That is, I put the device under test (DUT) in series with a 100Ω resistor, fed the output of the Bitscope Pocket Analyzer function generator into pair, and measured the RMS voltages across the DUT and across the resistor with a Fluke 8060A multimeter.
I then made a tank circuit like the one in the Colpitts oscillator, with the AIUR-06-221 inductor in parallel with a pair of the (nominally) 4.7µF capacitors in series. This should resonate at = 7kHz, with L=220µH and C=2.35µF.
It is certainly the case that the LC tank is resonating at the expected frequency, so now all I have to figure out is why the 180˚ phase changes that determine the oscillator frequency and that I measured with an external oscillator (in Colpitts LC oscillator) are not at or near the resonant frequency.