I don't have the Kindle app on my iPhone (the latest SE) but I have used the app iMazing to move other application files between my iPhone/iPad and my desktop Mac. I don't know how they do it, but it gets around a lot of the pain of moving data in and out.
The app runs on the desktop and recognizes an iDevice plugged into the desktop's USB port. I use it on a Mac but they have a Windows version also.
It isn't free but, to me, it has been worth the price.
I can't say for certain it will do what you want because I haven't done it, but it has made life very simple for moving other app data to and from an iDevice. Check out their website; maybe you can ask their tech support if it will work for you.
https://imazing.com/