SITUATION: I'm trying to connect my Lambda Function with my database in RDS. post had a very succinct solution. However, I'm stuck in one of the last parts of the solution where the SG of Lambda is supposed to be included in the inbound rule of RDS instance.
PROBLEM: The problem is that my lambda function is in one VPC (which has everything - NAT Gateway, Internet Gateway, Route Tables, Subnets, Security Groups - configured) while the RDS instance has always been in the default vpc (I didn't want to make changes to the default VPC). When I look for SGs while trying to add the inbound rule, I can see all the SGs in the default VPC but not in the custom VPC. Also, I don't see the desired SG when I try to "Set up Lambda Connection" in RDS - only those in the current VPC. Because of this, I get an error in Lambda that my task times out after 30 seconds.
ALREADY TRIED: I've established the Peering Connection between the 2 VPCs but still can't see the SGs. The problem still persists.
Here is a screenshot of the Route Table from Peering Connection:

IMO you need to reconsider having RDS in a VPC outside of the other VPC. This is not correct architecture, and leads to the access problem you are experiencing.
The post you reference is providing guidance in creating a new VPC, so as to not modify the default VPC. It doesn't suggest you should put your RDS instance in the default VPC, and your lambda in another. It suggests, single (new) VPC, but with all of the required resources in it.
PS Welcome, great first question. Touches on great key services and concepts.