AWS won't let you connect to ElastiCache instances from outside of their network even if you set up security groups to allow traffic from your IP address. All connections must originate from within their network.
Given this constraint, how can I test an application that heavily rely's on ElastiCache locally without creating a local instance?
There are many ways to connect to private resources in a VPC from outside of AWS. The most common one for testing and development purposes is through ssh tunnel as explained in AWS docs:
The other one, more for production deployment, is through a VPN between your home/work network and your VPC.