media_handle_upload for local files?

You want media_handle_sideload()

Handles a side-loaded file in the same way as an uploaded file is handled by media_handle_upload().

// Array similar to a $_FILES upload array.
$file_array = array(
    'name'     => 'filename.jpg',
    'tmp_name' => 'path/to/filename.jpg',
);

// Post ID to attach upload to, 0 for none.
$post_id = 0; 

$attachment_id = media_handle_sideload( $file_array, $post_id );