My https://datasette.io/ project is built around a similar idea to this: the original inspiration for it was Zeit Now (now Vercel) and my realization that SQLite read-only workloads are an amazing fit for serverless providers, since you don't need to pay for an additional database server - you can literally bundle your data with the rest of your application code (the SQLite database is just a binary file).
If you want to serve a SQLite API in an edge-closest-to-the-user manner for much larger database files (100MB+) I've had some success running Datasette on https://fly.io/ - which runs Docker containers in a geo-load-balanced manner. I have a plugin for Datasette that can publish database files directly to Fly: https://docs.datasette.io/en/stable/publish.html#publishing-...