Mosfet Switched capacitor in LC tank
I see two problems with that circuit:

1) as you wrote: the biasing of the MOSFET

2) The bulk diode of the MOSFET, when on the tank the voltage on the top node is negative relative to the bottom node, the source-bulk diode in the NMOS will turn on. Oops !

I have a proposal to solve both issues:







simulate this circuit &ndash; Schematic created using CircuitLab

The 1 M resistors bias the NMOSFET's Drains to ground, that solves 1) 

You cannot get rid of the source-body diode in discrete MOSFETs so I propose to use two in anti-series. Combined the two NMOS will behave as one NMOS without a body diode.

Instead of biasing the drains to ground you can bias them to any DC voltage as long as Vgs is changed accordingly. Such that there's a positive Vgs to turn the NMOS on and Vgs = 0 V to turn them off.

Note that this only allows you to switch on/off capacitance (C2 in series with C4) so the adjustment is coarse but you describe that in your question so you're aware of that.

If you want finer adjustment, place many of these sections with different values for C2 and C4 in parallel with the tank.