"Another problem has reared its head. After mixing only frequencies above 1kHz show themselves, i.e. from more than 10cm away the hand will change the frequency of the variable oscillator but this won't translate to beats being output by the mixer. The difference in frequency between the variable and reference oscillators has to be more than 1kHz for the mixer to give an output." -- innominata
This is oscillator coupling. If you can get the coupling frequency lower then you might use it productively to alter the bass timbre, though the far field will be less linear than it ideally could be. Or you can completely eliminate coupling and improve far field linearity.
You should decouple your oscillators via RC on the transistor power leads, like 10 ohms in series and 0.1uF or 1.0uF ceramic to ground.
If that doesn't do it, look at your mixer for coupling issues. You can add emitter followers to really kill coupling here.
Finally, it could be that your fixed oscillator LC impedance is too high and is picking up the variable oscillator C field. Try cutting the fixed L in half and doubling the fixed C to see if that improves things. If so, do it again. Keeping the oscillators physically apart helps a lot too.
"Then there are two design issues, that one and its enabler - the fact that the oscillator amplitude is changing with frequency. The latter I'm not sure how to avoid or whether it should be avoided. The former seems to be the primary design flaw."
A reduction in oscillator amplitude when the hand approaches the antenna is just a fact of life. The body isn't just the other plate of a perfect C, but has R too that damps oscillation. And just increasing C will also reduce amplitude.