Can you change your alternate apple id - somewhere? it is does not allow that on the current apple id screen.
You can change the alternate id only. Simply SIGN IN or go to appleid.apple.com and sign-in.
Once you sign in the 1st Tab will autonomously be selected for you : 'Name, ID and Email Addresses', if not select that one.
You will see a list of all the info related to that tab.
On the bottom you will see the alternate email address, below that section you will see the same thing but it will be editable. You can, add, delete, or edit any current additional emails to your primary apple ID email.
Again you can NOT alter your primary email address.