Vault secret access from outside EKS cluster

I have two aws account for Dev & staging. I have setup Vault pod without Agent injector on DEV AWS account on EKS cluster and expose the the vault address through ingress. I have setup Vault pod as well as vault-secret-operator and I am able to retrieve secret in Dev EKS cluster.

Now, I want to access same Dev account Vault on staging account. I have allow the staging NAT gateway IP in Dev vault’s ingress and I am able to telnet from a dummy nginx pod.

Now, need your help to retrieve the secret into Staging cluster. What is the best way to do that.
Through init container or some other way to do. Please suggest.

Example:
AWS Account Dev : dev-account
Vault address: https://devtest.example.com

above is example.

Please review my question once, I stuck in middle.