Also, just in case someone has tried it before. What's best practice for running a Typesense HA cluster on AWS? Has anyone been able to do it on ECS?

It is simple to just set up three nodes and list their IPs a 'nodes' file passed as argument while starting the service. Typesense will then take care of clustering. https://typesense.org/docs/0.20.0/guide/high-availability.html#configuring-a-typesense-cluster

Does this cover the use-case of horizontally scaling instances? I thought maybe I could freeze my dataset in an AWS instance and use auto-scaling to launch instances behind a load balancer

If your data is static then you don't need to use clustering. You can have standalone instances or use an ASG. But you wiill need to write to each standalone instance individually when you want to do a data refresh.