Minimum permission to execute sp_changedbowner SP
Community wiki answer:

From ALTER AUTHORIZATION (Transact-SQL):


  Requires TAKE OWNERSHIP permission on the entity. If the new owner is not the user that is executing this statement, also requires either, 1) IMPERSONATE permission on the new owner if it is a user or login; or 2) if the new owner is a role, membership in the role, or ALTER permission on the role; or 3) if the new owner is an application role, ALTER permission on the application role. 


There may be additional considerations in the AWS RDS world.

Note that sp_changedbowner is deprecated in favour of ALTER AUTHORIZATION. The sp_changedbowner documentation says:


  Requires TAKE OWNERSHIP permission on the database. If the new owner has a corresponding user in the database, requires IMPERSONATE permission on the login, otherwise requires CONTROL SERVER permission on the server.