From docker container need to connect RDS - Using Session Manager

I have installed AWS CLI and the session manager plugin inside the container and started the session. But is there any other way to make the connection from local and use the same connection to connect rds inside the container? FYI, I am using a rancher desktop.

1. Are you running your containers on a local machine or already on AWS? - My containers are running on my local machine using a rancher's desktop.
2. Is your RDS instance in a private subnet (no public IP, as in your previous question)? - Yes it is in the private subnet and has no public IP.
3. You're likely using the public Internet to connect to AWS? Neither VPN nor DirectConnect? - Starting SSM Session from my local machine using AWS account credentials to connect RDS. (SSM Session - EC2 - RDS)
4. Is anything not working with the AWS CLI or are you looking for an alternative? - From my local machine, If start an SSM session I can able to connect the RDS database. My question here is to start a session on the local machine and connect the RDS database from the container.

Currently, I am starting a session inside the container itself and in this case, I can connect the RDS database from the container.