The key here is Postgres 9.5. AWS DMS does not support it because they require logical replication support.
A few years back I migrated a PostgreSQL 9.2 database to AWS and wasn't able to use RDS because logical replication was not available.
I did try to use Bucardo but ultimately didn't trust myself to configure it such that it wouldn't lose data (first attempt left nearly all BLOBs unreplicated because the data isn't actually in the tables you set triggers on)
Physical replication to a self-built instance was easy, I was 100% confident it wouldn't be missing data, and the downtime from cutover was about 15 minutes (It involved restarting piles of slow Java applications)
A few years back I migrated a PostgreSQL 9.2 database to AWS and wasn't able to use RDS because logical replication was not available.
I did try to use Bucardo but ultimately didn't trust myself to configure it such that it wouldn't lose data (first attempt left nearly all BLOBs unreplicated because the data isn't actually in the tables you set triggers on)
Physical replication to a self-built instance was easy, I was 100% confident it wouldn't be missing data, and the downtime from cutover was about 15 minutes (It involved restarting piles of slow Java applications)