Superb ...
You have done a great job.
But one thing I want to clear that I have to derive tetrad using Kerr metric.
The method you above use ... it seems like you use tetrads to find ##g_{\theta\theta}## and other components of metric.
I also get relevant material in book named as
"Relativity_Demystified_-_a_self-teaching_guide"
Page no 100-101
But confused how to apply it to get tetrad comprising of four orthonormal vectors.