Measure voltage with no current
Neat physics solutions aside, the practical way to do this is with a very low input bias current op-amp running in a buffer configuration. One of these op-amps with a properly designed layout can draw down to single-digit femtoamps of current from your cap, making disturbances pretty much negligible, particularly if you only connect the amplifier to the cap when you're taking a measurement.

Analog legend Bob Pease describes measuring leakage of a polypropylene cap using this method:


  Now I will charge up some of my favorite low-leakage capacitors (such
  as Panasonic polypropylene 1 µF) up to 9.021 V dc (a random voltage)
  for an hour. I will read the VOUT with my favorite
  high-input-impedance unity-gain follower (LMC662, Ib about 0.003 pA)
  and buffer that into my favorite six-digit digital voltmeter (DVM)
  (Agilent/HP34401A) and monitor the VOUT once a day for several days.
  
  [...]

Day 0: 9.0214 V
Day 1: 9.01870 V
Day 2: 9.01756 V
Day 6: 9.0135 V
Day 7: 9.0123 V
Day 8: 9.01018 V
Day 9: 9.00941 V
Day 11: 9.00788 V
Day 12: 9.00544 V
Day 13: 9.00422 V

  
  The first day after soaking for an hour, their leak rate was as good
  as 2.7 mV per day. Not bad.


If you need to automate such a setup, a good old-fashioned reed relay has basically negligible leakage (better than even modern solid state analog switches) and can be used to briefly connect your amplifier to the capacitor under test in order to take a reading.