A common reason for that is using wrong PostgreSQL version. RDS proxy supports only some version:
For RDS PostgreSQL, RDS Proxy supports version 10.10 and higher minor versions, and version 11.5 and higher minor versions. For Aurora PostgreSQL, RDS Proxy supports version 10.11 and higher minor versions, and 11.6 and higher minor versions.
Thus, if you use newest version PostgreSQL 12, RDS proxy will not show it as its not supported.
You still can use this db from lambda, but have to associate your function with the VPC where RDS is and setup security groups so that function can connect to the db.
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