Disable crop when uploading custom-header

You could override the crop() method in wp-admin/js/image-edit.js so that it does nothing. That would disable all cropping, not just for the custom header image.

You could remove the default control and replace it with your own that does a straight image upload without going through the media library.

I don’t see a way to prevent cropping only for custom header images while still using the media library. That does not mean there is not a way. I just am not seeing any.