<?php
// $old_value is just for testing purposes, and should be removed.
function wpse315119_meta_trick( $filter, $object_id, $meta_key, $meta_value, $unique_or_prev_value, $old_value = null ) {
    // Remove the filters and save the new meta value. Make sure that
    // the priority and number of arguments are exactly the same as
    // when you added the filters.
    remove_filter('add_post_metadata', 'wpse22728_meta', 20, 5);
    remove_filter('update_post_metadata', 'wpse22728_meta', 20, 5);

    // Manually save the meta data.
    if ( 'add_post_metadata' === $filter ) {
        add_metadata( 'post', $object_id, $meta_key, $meta_value, $unique_or_prev_value );
        echo 'Meta data added. Value is "' . $meta_value . '" and not "' . $old_value . '".<br>'; // test
    }
    elseif ( 'update_post_metadata' === $filter ) {
        update_metadata( 'post', $object_id, $meta_key, $meta_value, $unique_or_prev_value );
        echo 'Meta data updated. Value is "' . $meta_value . '" and not "' . $old_value . '".<br>'; // test
    }

    // Finally, re-add the filters.
    add_filter('add_post_metadata', 'wpse22728_meta', 20, 5);
    add_filter('update_post_metadata', 'wpse22728_meta', 20, 5);
}

function wpse22728_meta($check, $object_id, $meta_key, $meta_value, $unique_or_prev_value) {
    $the_filter = current_filter(); // keep this and don't change
    if( get_post_type($object_id) == 'mycpt' ) {
        if( $meta_key == 'mycustomkey' ) {
            $old_value = $meta_value;

            // Change the value.
            $meta_value = 'new_value';

            // Save the new value.
            wpse315119_meta_trick( $the_filter, $object_id, $meta_key, $meta_value, $unique_or_prev_value, $old_value );

            // Finally, set $check to the saved value, so that the $old_value won't be saved.
            $check = $meta_value;
        }
    }
    return $check;
}

add_filter('add_post_metadata', 'wpse22728_meta', 20, 5);
add_filter('update_post_metadata', 'wpse22728_meta', 20, 5);