What should I do if I encounter difficulties with launching the game from the account with the restricted rights?

You can try to add xpromo.g5.com in the list of unrestricted websites. 
You can follow the instruction below:

• Go to System Preferences.
• Select Parental Controls.
• Sign in the username and password.
• Click on "Website Restrictions".
• Select "Allow access to only these websites".
• Add xpromo.g5.com to the list.
• Press "OK" and start the game once again.