This is strange, because the synchronized birthday calendar is just a calendar like all others, and I don’t think the Contacts app or what else should put birthdays into randomly selected DAVdroid calendars.
Are you sure that you didn’t try to change the birthday in the Calendar app? (But it shouldn’t be possible anyway, because the calendar is falgged read-only.)
I’ll mark this as solved. If you find out something relevant or if there’s anything I can do for you in this matter, just let me know here.
PS: https://github.com/nextcloud/server/issues/11633 – 403 instead of 404 could help to understand the problem when someone encounters the error message