Override custom-header.php

Well, I figured it out, if anyone is interested. Not sure if this is the proper way of doing it, but this is what worked for me.

You do not need a custom-header.php in your child theme. The first thing to do is to remove the action that defines the custom header attributes in custom-header.php. Create a child theme functions.php and put this in it:

// remove default action from simone
remove_action( 'after_setup_theme', 'simone_custom_header_setup' );

Next, copy the function you want to change from custom-header.php and paste it into the child’s functions.php. Then change whatever attribute you want and add the action back (make sure to use your child theme’s name in the appropriate places). This is what it looked like for me:

// define new custom header size
function yourchildthemename_custom_header_setup() {
	add_theme_support( 'custom-header', apply_filters( 'simone_custom_header_args', array(
		'default-image'          => '',
		'default-text-color'     => 'ffffff',
		'width'                  => 1800,
		'height'                 => 1200,
		'flex-height'            => true,
		'wp-head-callback'       => 'simone_header_style',
		'admin-head-callback'    => 'simone_admin_header_style',
		'admin-preview-callback' => 'simone_admin_header_image',
	) ) );
}
add_action( 'after_setup_theme', 'yourchildthemename_custom_header_setup' );

In my case, I had to add one additional thing to my css to make my image full width:

.header-image img {
  width: 100%;
}