How to connect an EC2 in a VPC to an Aurora RDS in another VPC using VPC Peering

I have created an EC2 in VPC A I already have an Aurora RDS instance in VPC B I have successfully created VPC peering and verified it by pinging an EC2 instance in VPC B from the EC2 instance in VPC A.

Whenever I SSH into the EC2 in VPC A and try to connect to the RDS instance in VPC B using the MySQL client through the terminal. In case I allow all the traffic in the inbound rules from anywhere in the security group associated with the RDS instance -> the connection is successful In case I allow all the traffic in the inbound rule from only the CIDR Block of VPC A in the Security group associated with the RDS instance --> the connection is failing.

Notes:

• Ec2 instance in VPC A doesn't have role associated with it
• I found some answers that suggests to set the RDS instance private , so that it resolves to private IP. i.e. associating the RDS instance with a private subnet that has no Internet gateway , but how could a backend then connect to this RDS instance with no hostname in case we set the RDS to private ?