I have been playing with this for a couple of years and multiple Apple headphones. I have not been able to get Kindle to work on AirPods Max, AirPods 4 or AirPods Pro 2 (USB-C or Lightning). Interestingly enough it works on the older Apple headphones I have, specifically the Beats Flex and the AirPods Pro 1. The only theory I have is that all of the devices it doesn't work with are on newer Bluetooth standards, notably 5.3. All my Bluetooth 4.0 headphones work fine as do all my bluetooth devices from other brands.
I was hopeful that the newer generations of Kindle would fix this issue, but it still persists on the 2024 models. I have been unable to find the Bluetooth specs on the new Kindles to follow up more on this theory.