the file placed in the child theme is not included

You want to replace bimber_capture_entry_featured_media in your child theme. In the theme.php file you’ve given us you can see that this is only called in one place, from bimber_render_entry_featured_media, and the call is wrapped in a filter:

if ( apply_filters( 'bimber_render_entry_featured_media', true, $args ) ) {
    echo bimber_capture_entry_featured_media( $args );
}

where returning false from the filter will cancel the call to the default code. Assuming that’s true, that nothing else calls this function without the filter guard, what you can do then instead is

1. implement the bimber_render_entry_featured_media filter; choose a low priority if there’s anything else that implements this, but I don’t see anything in these files
2. in the filter call your replacement bimber_capture_entry_featured_media_my_modifications and echo the result
3. return false from the filter to cancel the call to the original unmodified code.

e.g. (untested)

function bimber_render_entry_featured_media_filter_my_modifications( $flag, $args ) {
    if ($flag) {
        echo bimber_capture_entry_featured_media_my_modifications( $args );
    }
    return false;
}
add_filter( 'bimber_render_entry_featured_media',
            'bimber_render_entry_featured_media_filter_my_modifications', 10, 2 );

However there are plenty of hooks in the original method too, so depending on what you’re trying to do you might be able to use those to change the behaviour rather than completely overriding the method.