Issue using a single lambda to access RDS resource on VPC and access internet

I have an RDS on a VPC with a security group and 2 subnets. I have a lambda that needs to query the RDS and then upload this information to a third party via the internet. When I try to use this lambda without attaching the security group / subnets, I can access the internet, but not the RDS. When I attach the security group / subnets, I can access the RDS, but not the internet. The security group allows all traffic both inbound and outbound. What are the exact steps do I need to take to accomplish my objective? I'm trying to go for a full serverless approach, and this function only needs to run once per day (for around ~10 seconds), so I'd like to not have to pay for a server that needs to run 24/7 to accomplish this objective.