Is there a way to experimentally separate the effects of gate voltage?
Not sure what kind of leakage you are concerned with. If it's gate leakage, you can measure that directly (at least at DC and room temperature). 5pA is not all that small. A (relatively) inexpensive op-amp such as the LMC6001 has a guaranteed maximum bias current of 25fA which is 200 times lower than your estimated leakage. 

You need to take precautions to get that performance, and follow recommendations from the manufacturer (National was the original source) and gurus such as Pease. 

You can also buy an electrometer type instrument (as suggested in the comments), and that will take care of some of the problems and promise higher performance, but there will still be recommendations to follow to measure currents like that without excessive error. 

If you have to make AC measurements it may get much trickier.