Pretty urls for custom pagination

You can handle this with WordPress’s internal rewrites system by adding a rewrite endpoint:

function wpd_cpage_endpoint() {
    add_rewrite_endpoint( 'cpage', EP_PERMALINK );
}
add_action( 'init', 'wpd_cpage_endpoint' );

This will add extra rewrite rules enabling /cpage/n/ on the end of permalinks.

You can then fetch the value on those requests with get_query_var( 'cpage' ).

EDIT-

This is the code I used to register the post type and endpoint:

function wpd_test_post_type() {

    $args = array(
        'label' => 'Test Type',
        'public' => true,
        'rewrite' => true,
        'supports' => array( 'title' ),
    );

    register_post_type(
        'test_type',
        $args
    );

    add_rewrite_endpoint( 'cpage', EP_PERMALINK );

}
add_action( 'init','wpd_test_post_type' );