What you are trying to do in effect is modify the system tables underneath the sys.database_principals catalog view which is disallowed when the database is read-only. Instead you'll need to align the Server Principal with the Database Principal by dropping the Server Login on the secondary instance and recreating it using the same SID that was used on the primary instance.

How to transfer logins and passwords between instances of SQL Server

PS You will no longer need sp_change_users_login for this scenario but for future needs please consider switching to use ALTER USER instead. sp_change_users_login was marked obsolete in SQL Server 2008 and will be removed in a future version of the product.