First of all I would strictly differentiate between the **point estimate** of the Cpk value and the associate **uncertainty**. There is nothing inherently wrong with using only a few measurements to calculate a Cpk value, however, the uncertainty of this estimate is probably large. In addition, you should make sure that all parties (you and your costumer) use the same formula to calculate the Cpk estimate. Different methods exists, because the Cpk value uses the standard deviation of the population, and you have to agree on an unbiased estimator -- the formula, which is used for the Cpk value.

Using multiple measurements per part ensures that your data analysis becomes complicates: Assuming that the measurements "within" a single part are correlated, you either have to use advanced data analysis, or you have to find a way to get rid of four data points. The reason for this is that the data points are not independent -- in the statistical literature you have performed "pseudo sampling" also called "sub sampling".

Although I understand that you want to make sure that the part has an homogenous thickness and no point lies outside the tolerance range, you should decide how to condense these five points into a single number, which incorporates the information of interest. E.g. if all five measurements are practically the same number, you could keep only the first data point -- it basically contains all the information. Instead, if you generated inhomogenous parts you probability should use the maximal value of the absolute deviation, max{ | zMeasured - zTarget| } for each part. Note, that due to the absolute value the tolerance is now single sided.

Personally, I get the impression that you and your customer did not properly specify the expected outcome of the study. Thus, I expect that your customer is not firm in statistics. In my experience it is not helpful to discuss statistics with such a counter part, because they are neither able to decide what is needed nor able to select a practical method to achieve this goal. Hence, I suggest that you first perform the calculation describes above -- it will take only 20min -- and then talk to the customer. If you communicate the short comings and uncertainty of the data analysis you have a high chance that the customer is already satisfied. Especially if you send the data and offer that the customer could perform some data analysis themselves.