Change register form action url

You could do something like this:

<?php

add_action( 'register_form', function() {
    $get_type = filter_input( INPUT_GET, 'type', FILTER_SANITIZE_STRING );

    if ( ! $get_type ) {
        return;
    }
?>
<input type="hidden" name="type" value="<?php echo esc_attr( $get_type ); ?>" />
<?php
});

This will add the type parameter as a hidden field on the form if it is set in the GET query string.