FM demodulator with center frequency tracking

So I have a capacitive vibration sensor, it drives Vackar oscillator frequency, but there's a problem: there's a large, low-frequency (<15 Hz) "common mode" capacitance that can shift the center frequency between 500-900 MHz. What are the best ways to demodulate the signal (5 MHz max deviation)? This definitely has something to do with AFC but I'm having trouble building a wideband frequency detector that will track the center frequency. I tried making a quadrature detector with low-Q tank but it loads input amplifier so heavily that the output amplitude drops to unacceptable levels. What am I doing wrong? I followed this paper to design it. Here's the schematic of input limiting amplifier and quadrature detector.

what I would do would be to have a phase locked loop with a narrow loop bandwidth, and look at the oscillator control voltage when it is locked to determine the frequency. That way you now have a phase locked oscillator that is tracking the center frequency of the device under test.

I don't quite understand, how this setup will work. If loop bandwidth is narrow then the capture range is narrow and if initial signal frequency is outside the capture range, the PLL will not lock, right?

But capturing a carrier that is swinging between 500-900 MHz with a 15Hz deviation frequency is pretty difficult.It looks like frequency hopping radios but a encrypted frequency information is transmitted while digital data is sent.

Now I've got 2 ideas: PLL with adjustable bandwidth filter (switches between wide and narrow) or reduce oscillator frequency. Second way will reduce oscillator' sensitivity to capacitance but the drift bandwidth will also reduce. What do you think?