How to add a calculated field to Members database AFTER successful login

Have you actually read up about the Wix Members app PrivateMembersData collection?
https://support.wix.com/en/article/corvid-wix-members-privatemembersdata-collection-fields

Permissions

The PrivateMembersData collection has the following permissions :

  • Read: Site member author

  • Create: None

  • Update: None

  • Delete: None

You cannot change the PrivateMembersData collection permissions.

To do something like that you will need to create a separate dataset, you can see a good example here for making a member profile and update profile pages.
https://support.wix.com/en/article/corvid-tutorial-building-your-own-members-area

Now obviously you do not have to keep the tutorial and all on it exactly the same as it is, you can swap and change it to suit whatever needs you want for it.