Add featured image below content with priority

This should be very doable. I’d start by ensuring that the plugin always uses the large image, not the backstretch:

add_filter( 'display_featured_image_genesis_use_large_image', 'prefix_use_large_image' );
function prefix_use_large_image( $post_types ) {
	return get_post_types();
}

Then you can use this filter change which hook the plugin uses:

add_filter( 'display_featured_image_genesis_move_large_image', 'prefix_move_image_to_content' );
function prefix_move_image_to_content( $hook ) {
	return 'genesis_entry_content';
}

This filter can modify the priority on the hook (I’m guessing on the number, not knowing where the sharing buttons are added, so you may need to experiment):

add_filter( 'display_featured_image_genesis_move_large_image_priority', 'prefix_change_image_priority' );
function prefix_change_image_priority( $priority ) {
	return 20;
}

Hope that helps get you started–

Thanks Robin for the very quick answer!
That’s very helpful – I’m almost there. Will carry on tinkering.