How to take resistance reading when value is effected by measurement?
You're trying to measure to within a couple hundred ppm. Unless it's very expensive, I doubt your LCR meter is really anywhere near that accurate or stable. They're not designed to take precise DC measurements. 

I suggest using a decent benchtop multimeter. My old Agilent 34401A uses a 5uA current source on the 1M\$\Omega\$ range. It should be stable in the short term to a few ppm, which would allow a measurement accuracy in the 1% range if the pot resistance stays put. 

Temperature changes are a possibly limiting effect, but more likely to be a serious issue if your pot is a carbon type. Typical tempco of a conductive plastic pot might be in the 100ppm/°C range, so you have to make sure it stays relatively constant during the measurement. 



The 5uA causes heating of I2R = 12uW and your 0.5V RMS causes heating of 0.5uW. Neither one is very large, but the LCR meter is considerably less. If the tempco of your DUT is really bad you may need to make some kind of custom measurement apparatus. Unfortunately, I don't see any way to keep the dissipation constant and make the necessary measurements so it might involve switching tiny currents very accurately and/or perhaps immersing the element in something like oil. 

Interesting problem!