Set Display Name on Membership Checkout and for BuddyPress Name field.

my_first_last_display_name.php

<?php
/**
 * Set Display Name on Membership Checkout and for BuddyPress Name field. 
 */
function my_first_last_display_name( $user_id, $morder ) {
    
    // Get user's first and last name.
    $first_name = get_user_meta( $user_id, 'first_name', true );
    
   if ( ! empty( $first_name ) ) {
    	$display_name = trim( $first_name );
	} 
	
	if ( ! empty( $display_name ) ) {
		// Should set "display_name" as well as the BuddyPress Profile field name.    
		$args = array(
			'ID' => $user_id,
			'display_name' => $display_name,
		);
		// Update WP user display name.
		wp_update_user( $args );
		
		// Update the 'Name' xprofile Field (field ID 1).
		if ( function_exists( 'xprofile_set_field_data' ) ) {
			xprofile_set_field_data( 1, $user_id, $display_name );
		}
	}
}
add_action( 'pmpro_after_checkout', 'my_first_last_display_name', 20, 2);