How to transfer an Amazon S3 bucket to another account?

If you are going to do this frequently then you should create a new account per customer and then transfer ownership of the whole account to the client. See Consolidated billing and Organisations.

S3 Buckets cannot be transferred between accounts. At least in the simple sense of "here is my bucket, now it is your bucket". Everyone seems to use some form of copying. If you have permission to both your original bucket and their destination bucket then you can use the AWS CLI and just

aws s3 sync s3://bucket1 s3://bucket2

Have you tried adding their account as an ALL PERMISSION user to one of your buckets? http://docs.aws.amazon.com/IAM/latest/UserGuide/roles-creatingrole-policyexamples.html

Then login as their account and see if they can then edit the policy to remove your original account? Not sure how the billing would turn out since you created the bucket.