Thank you all for your informative replies.

Turns out I had two problems. One was that my scope probe capacitance
(30pF) was loading the oscillation. However, the spectrum analyzer is
sensitive enough though that if I simply place the probe close to the
oscillator, the radiated signal from the oscillator is picked up.

The second problem was my bench, for I tried the antenna probe trick
before without success. I had the solder joints on my board touching
the ESD protection mat that was acting like a semi-conductive surface I
think. Once I placed the board on spacers the issue resolved itself.

I now get about 2kHz of tuning over 9V with 2.8dB of amplitude change
(this includes a slightly non-linear region of the varactor in the last
2-3 volts). Selecting an 850 Hz interval and applying a 100 Hz
modulating square wave produces a beautifully symmetric FSK spectrum.

Cheers,
Chris