Fetch and set the value for the display name user field in admin.

pmpro-set-default-value-for-display-name-field.php

<?php
/**
 * Set the default value for the display_name field when editing a user in the WordPress admin.
 *
 * This recipe assumes a PMPro User field for the display_name exists.
 *
 * You can add this recipe to your site by creating a custom plugin
 * or using the Code Snippets plugin available for free in the WordPress repository.
 * Read this companion article for step-by-step directions on either method.
 * https://www.paidmembershipspro.com/create-a-plugin-for-pmpro-customizations/
 */
// Set the default value for the display_name field.
function pmpro_set_default_value_for_display_name_field( $field, $where ) {
	// Only set the default value for the display_name field in the admin.
	if ( is_admin() && current_user_can( 'edit_users' ) ) {

		// Skip if we are not on a page where a user's ID is passed?
		if ( ! isset( $_GET['user_id'] ) || empty( $_GET['user_id'] ) ) {
			return $field;
		}

		// get the user's display name
		if ( 'display_name' === $field->name ) {
			$user_id   = intval( $_GET['user_id'] );
			$user_data = get_userdata( $user_id );
			if ( $user_data ) {
				$display_name = $user_data->display_name;
				$field->value = $display_name;
			}
		}
	}

	return $field;
}
add_filter( 'pmpro_add_user_field', 'pmpro_set_default_value_for_display_name_field', 10, 2 );