I attacked the same problem somewhat differently and found some interesting groups.
You can switch the exponential map you use, use instead a Mittag-Leffler function and find a whole family of groups going from sum to multiplication.
Being groups you can define generalized Fourier/Mellin transforms with them.
I wrote a blog post explaining the approach (there is a link to the paper at the end for more details).
https://paureahack.blogspot.com/2016/06/supersum-subproduct....