Phase noise reduction in LC oscillator with a high-loss capacitor

XY problem, you have what is effectively a capacitor microphone element, with one side grounded and you wish to build an RF Mic.

Were it me, I would not make it part of the oscillator instead using a crystal osc, and putting the transducer in a bridge circuit. Use a couple of fixed inductors and a piston trimmer or such (shunt it with a variable capacitance diode so you can servo the operating point to compensate temperature drift), then do phase detection on the bridge output voltage.

A couple of inductors, a piston trimmer, a varicap diode, a crystal osc, and some sort of phase detector (maybe a diode ring mixer?) followed by a lowpass filter and some opamp bullshit, job done.

The HP3033 synthesizer used a VCO something like this

^{simulate this circuit – Schematic created using CircuitLab}

Notice how the two VDD bypass capacitors have their RETURN nodes tied into the resonator RETURN, for quiet operation. Also, C5 is part of the resonant path, thus its AC current is affecting phasenoise.

You might install a Schottky diodes across R2, if you want a fixed Vout amplitude of 0.3v PP.

You should make C3 quite small, so only a portion of the resonant energy is dissipated in that base resistor.