Have you tried the method of hooking a known resistance in series with the pickup, driving the circuit from the sine wave generator and adjusting the frequency until the voltage across the resistor is the same as that across the pickup? At that point, the impedance is the same as the resistor value, so you can solve for the inductance. Might be tedious unless you have 2 voltmeters though. You could always borrow my DMM if you need a second meter.

It might be interesting to see if this method gives values close to the parallel cap method. You could also test with different resistor values. I suspect that every test will give a slightly different result, so maybe just take the mean of all of them?

DC