AWS Lambda in VPC with RDS and Internet Connection

Question

I set up an Aurora Database in a newly created VPC and no public accessibility. As I want to run a Lambda function in the VPC that is able to both, access the RDS instances as well as the Internet, I changed the routing tables of the RDS instances to allow traffic from a NAT gateway which I placed in a public subnet in the same VPC.

For the Lambda function itself, I created a separate private subnet, also just allowing traffic from the NAT gateway in the routing table. I assigned this subnet and VPC to the Lambda function in the Lambda settings. The internet connection works fine with this configuration but I can not access the database. That's why I followed this post (https://serverfault.com/questions/941886/connect-an-aws-lambda-function-triggered-by-api-gateway-to-aurora-serverless-mys) and added the IP CIDR of the Lambda subnet to the Security Group of the RDS instances 

Still, the Lambda function is able to interact with the public internet but can not connect to the RDS instances (timeout). I'm running out of ideas, what is wrong here?