SPC question for batch style processing

Hello Everyone...I have a question in regard to batch style processing and how that affects SPC. I run a lapping and fine grinding shop and we specialize in tight tolerance work. When we process parts we run in batches, for example 6 carriers of parts in a load with 10 parts per carrier, 60 parts per load. When we do this our variation within a load is extremely low, most times less than .00003". Where we see variation in our process it is from load to load, never within a load.

Our SPC software is setup in subgroups. When we inspect and document parts we inspect one part out of each carrier and plug those readings into a 6 piece subgroup in our SPC software. As mentioned, the variation that we see is from load to load as opposed to within a load, so this gave us better statistical values. I'm having issues with a customer that disagrees with this method, he is doing random sampling of finished parts and as expected, they are getting lower values than we are when it comes to Cpk. That part I understand, but what I don't understand is that their values are very low compared to ours.

The parts in question have a .0004" total thickness tolerance. We ran 1,200 parts as a sample to prove capability, (20 loads) and only had .0001" of variation between the lowest parts and the highest parts that we inspected, which was 120 pieces inspected and documented. We used just less than 1/4 of the tolerance. When my customer did their random inspection they agreed with our readings, but they show a Cpk value of 1.02. I don't understand how that is possible considering we only used 1/4 of the print tolerance and held 1,200 parts consistent within .0001" of each other. They require a 1.33 Cpk on these parts and from what I'm seeing we would have to hold .00005" total or 1/8th of the print tolerance in order to meet those type of Cpk values using their methodology. Something isn't adding up.