I might be wrong, but as far as I remember static catalogs cannot be searched using pystac_client (though there might be a different project that helps you do that).
I’m certainly no expert, but what I eventually ended up doing for experimentation purposes (deployment is a different issue, but I don’t have any experience with that) is use stac-fastapi’s docker-compose file to spin up a suitable PostgreSQL database and a STAC endpoint, and use pypgstac’s command line tool to ingest the static catalog.