@Tanul Thank you for your patience over the matter and for all the inconvenience this issue might have caused.
One such article which demonstrates configuration of the nginx ingress controller via a ConfigMap to pass a custom list of headers to the upstream server is as below:
https://kubernetes.github.io/ingress-nginx/examples/customization/custom-headers/
This article might not suffice to your requirement but still might be helpful in understanding custom headers role in nginx.
Keyvault integration with AKS recently came under Supported Scenario and there are still scenarios which Product Group is working on so you can add your suggestion here on our feedback forum as well where our Product Group is quiet active and post with highest number of upvotes is taken into consideration and worked upon.
I have provided your input to our internal team as well to take into consideration going forward and if there are any updates on that part I will keep you posted on same.
Hope it helps!!!
If the suggested response helped you resolve your issue, do click on "Mark as Answer" and "Up-Vote" for the answer that helped you for benefit of the community.
Note : The articles in the mentioned answer which are not the official documentation from Microsoft side, so we can't confirm on the working of components as mentioned in the article. We have just tried to provide you the information which can be helpful to you with the current issue you are facing and help you in resolving the same.