Hi Erik
Welcome to the real word (it's a joke),
Most of what is written take "normal distribution" (gausian) in account, even mean it self is calculated with the same asumption.
It's for shake of simplicity, but in many cases is a good aproximation, I agree with you about non normality and it's my experience that, as Donald Wheeler wrote; "the central limit theorem has nothing to do with control limits", because the non normality of the range chart, and that the control limits are calculated with them.
Many practitioners will recommend to normalize data, but as the same autor said, most of the time, the chart loss the capability to be understand easily.
I readed an article that say that box-cox transformation is the best
"Computing Process Capability Indices for Non-Normal data: a Review and comparative study", by Loon Ching Tang and Su Ee Than; in the Quality and Reliability Enginnering International. Int 15: 339-353 (1999)
That article compares Wright's index; Clements' method; Box Cox transformation; and Johnson transformation.
Or use the "Yeo-Johnson Power Transformations" by Sanford Weisberg of the departament of Applied Statistics, University of Minnesota in Oct 26 2001
Or check for
"A capability Index for all Occasions" by K.S. Krishnamoorthi and Suraj Khatwani, that was exoposed in the ASQ's 54th Annual Quality Congress Proceedings; ussing a Weibull model.
You can still depend on non parametrical Capability index (also shown in the same article)
Cpu = (USL - median) / (X0.99865 - median)
Cpl = (median - LSL) ( (median - X0.00135)
You can still try to estimate variation to each side of the median (it's not written anywhere but I use it), and use the estimate to calculate the index.