You must be having a scheduled job which does database restoration from Production to your server meaning you will have specified time and you would also be knowing based on historical data that how much time does this process take. you may create another scheduled task inside SQL Server as SQL Job or Windows scheduled task which would include user permission script as well as other script.
You will have only one risk - if restoration fails from production then you can't set dependency to this job as the first one is triggered from third party application. In any case, granting permission has no consequence however don't know your script, it might have some consequences.
This would reduce overhead for you and it will become part of your scheduled job and when you start working, all is set for you every day. I hope this helps.