You did not mention at what frequency you want to operate. To measure varicap diode capacitance vers. voltage,I would build an oscillator running at the frequency of interest, and use a variable tuning capacitor to tune it. This capacitor can be measured on a LCR meter, or, estimated from the visible mechanical parameters. The I will replace this capacitor with the varicap diode, and record the frequency as a function of voltage.
Some LCR bridges apply unknown DC or AC voltages across capacitors under test, and can damage the diode. As you plan to use the diode in an oscillator, the described procedure is the best way to measure the diode.