Fundamentally, it seems that the methodology could integrate into the Improve or Control phases of DMAIC. Establishing a continuous improvement culture with the 4 step Improvement Kata could certainly lend itself to sustaining solutions and driving further improvement.
Some additional questions come to mind:
1) What might be the scope of the Kata usage within a DMAIC project?
2) What role would the Belt serve? Setting up the system or performing the role as Coach?
3) What level Belt might best play a role? Black, Green, or Yellow?
4) Or would this be best left to the Process Owner to run and execute post project closure?
The beauty of Six Sigma is in the rigor of following the DMAIC framework, yet having the flexibility to use the right tool for the right job.