I raised a similar question recently here which was resolved, however I've change the setup slightly.
The Nat Gateway was costing a small fortune so I switched it out for a Nat Instance using EC2. I followed this guide.
Now my Lambda is successfully connecting to both the RDS instance and also has internet connectivity which is great. The problem I now have is I'm unable to connect to RDS Postgres through port 5432 from my local machine and I'm a bit stumped.
Screenshots here, let me know if you need anything else.
Cheers!