AWS Lambda in VPC with RDS and Internet Connection

Question

I set up an Aurora Database (provisioned) 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 (called rds-launch-wizard).

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?