Is there an algorithm to generate the Kawase Blur Kernels to approximate the actual Gaussian Blur (of a specific kernel size, and if possible, a sigma)?
Looking at Intel - An Investigation of Fast Real Time GPU Based Image Blur Algorithms By Filip Strugar it looks like the Kawase kernel is just a way of implementing a linear kernel quickly, but in a way that constrains the kernel somewhat.
This means that you could make such an algorithm. Either choose a set of spreads and adjust their weights (if that is possible) for a best fit, or search a number of possible spreads (with the best weights) and find the ones that fit best.
Given the ad-hoc nature of the kernel I suspect you'll end up with a table of best fits for each set of spreads, and then you (or anyone using the table) would want to make a judgement call about effect vs. clock ticks.
The kernel is designed with some specific hardware in mind, so you'll need to check to see if you can adjust the weights at all, or if you can only adjust the spacing and the repetition at each spacing. At any rate, because the spacing is discrete, its a fairly highly nonlinear optimization problem. It may be best to just do a brute-force search at each size of Gaussian you want to emulate. Just try out a bunch of different spacing combinations, and see which ones give you the closest match to a Gaussian of a certain size for the number of iterations.