However, FusionAuth doesn't currently support the AWS signature for Elasticsearch requests.

The recommended way of securing such clusters is to place it in a private subnet and restricting traffic to it using a security group. More information: https://fusionauth.io/docs/v1/tech/installation-guide/securing#fusionauth-search

If you have to make it public to make it accessible to resources outside if AWS you could use a source IP lock, a VPN, basic auth if AWS supports it, or you could proxy the request perhaps to another endpoint that can build the AWS sig v4 header.