Is there a clever way to use a small variable capacitor where a larger capacitance is required?
You could try an op-amp capacitance multiplier circuit:


(Courtesy of Wikipedia)

C = C1 * R1 / R2, so adjust R1 / R2 to provide the level of multiplication desired. Note that some op-amps have trouble driving capacitive loads. So if you connect a significant-enough capacitance to the op-amp output, it may oscillate.

Edit: A variant to try is this one. It uses the LM102 to buffer the input current, mostly eliminating the "resistance factor" from the previous circuit. I have not tested it, but would be skeptical of it's performance with very small capacitances. Also note that since C1 is being driven by the LM101A, this is "capacitive loading" and some op-amps could oscillate in this configuration. Some experimentation would be required. (Courtesy Texas Instruments)