Hey bfvassum,
We'd be happy to help out. When you add an iCloud.com email address to your Apple ID, this won't replace whatever you currently have set as the primary email address on file or change your Apple ID. It adds it as a secondary and alternate login only. The only way to set up an Apple ID as an iCloud.com email address is the primary is to create a whole new Apple ID and not link an existing email address to the account.
Since all your purchases and information are associated with your current Apple ID though, we'd suggest just setting an iCloud.com email alias, not setting up an entirely new Apple ID. You can still use the iCloud.com email address for logging in, and providing to others, but the Tutanota would remain the primary email address on file unless you change it to something else.
For more info, please review these steps:
Create an email alias
1. In Mail on iCloud.com, click 
at the top of the Mailboxes list, then choose Preferences.
2. Click Accounts, then click “Add an alias.”
3. Provide the requested information:
• Alias: The text you provide becomes the email address (alias@icloud.com). An alias must contain between 3 and 20 characters.
• Full Name: The name you provide appears in the From field of the messages you send.
• Label: Aliases are listed alphabetically by label.
4. Click Add, then click Done.
All this and more is available here: Add and manage email aliases for iCloud Mail on iCloud.com - Apple Support
We hope this helps out with better understanding your options.
Take care!
