Tank circuit oscillator design

First, you would have to accurately characterize the frequency shift over temperature, then arrange to adjust the value of some other component (presumably the capacitor) in such a way that the change in inductance is balanced out to maintain a constant frequency.

Not an easy (or inexpensive) task.

Instead of using the inductor as a frequency-determining element in an oscillator, generate a 10 kHz signal through some other means (e.g., a 555 timer astable circuit, or a crystal oscillator and divider) and drive the inductor with that signal through some sort of driver circuit that gives a constant output amplitude.

What you are describing is similar, but not identical, to commercial electrodeless conductivity sensors which use a pair of toroidal inductors (one drive coil and one receive coil) coupled to one another by the conductivity of the fluid to be measured. This Rosemount whitepaper gives details; see Fig. 5 on page 3.

I'm not sure what kind of results your method would yield.

 

Why does the frequency matter?
If a transmitter torroid and similar receiver torroid are at the same temperature then they will track each other.