2.2 uH might work with a circuit that precisely measures the inductor loss tangent respective real impedance part, but rather unlikely with the simple methods you are using up to now. In metallurgy, there are also eddy current measurement methods utilizing pulse response.
As another problem, coil resistance becomes a relevant part of the loss in this setup, so you get stronger temperature dependency with the small coil. I would rather go for a usual inductive sensor geometry with a ferrite pot core half.