Perhaps you could have a local/internal Craft install and then regularly run a script that crawls the site and saves all the pages as static HTML. Tools like HTTrack already do a pretty good job of this.
Or there are static cache plugins like Blitz that you might be able to utilise. They’re designed for caching, but if you warm the cache you might be able to export the static files for uploading to your host.
UPDATE: Blitz 3 makes this possible. Read the announcement at: https://putyourlightson.com/articles/blitz-3-embracing-the-jamstack