Migrating RDS PostgreSQL to Aurora PostgreSQL in a different account

Look into Amazon Aurora Fast Cloning as well. It supports Cross-account cloning via Resource Access Manager:

• https://aws.amazon.com/about-aws/whats-new/2019/07/amazon_aurora_supportscloningacrossawsaccounts-/
• https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide/Aurora.Managing.Clone.html#Aurora.Managing.Clone.Cross-Account

Possible route to take, this will have downtime but will still be minimal, and doesn't require DMS or manually dealing with logical replication:

• Add an Aurora replica to the RDS pgsql DB instance on the same source account: https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide/AuroraPostgreSQL.Migrating.html#AuroraPostgreSQL.Migrating.RDSPostgreSQL.Replica
• do additional testing on Aurora replica on this source account to check for any compatibility issues
• At time of cutover, create a clone of the Aurora DB cluster that will allow the clone to be created in the target account
• perform DNS cutover to the cloned aurora DB cluster
• delete/terminate source account Aurora DB cluster when appropriate, entirety of the storage will be billed now towards target account

Having said that, to err on the safe side, I personally would recommend customer to separate this into 2 tasks and not try to group into one operation. One to upgrade to Aurora, and the second to create the fast DB clone cross-account. It will require 2 DNS cutover for the applications or require 2 updates in route 53 if you're using CNAMEs, but will likely make things clearer.