I'm mostly focusing on sqlite-as-the-data-source, but I can imagine something like this might be in scope for a project like lunr.js[1], which currently uses a large index served as JS/JSON. You can write a Jekyll plugin in a few lines of ruby code (with no gem management needed, just drop it in the _plugins directory), so the "write-index-to-a-sqlite-file" part shouldn't be hard to build either.
Is FTS5 included in the SQLite browser builds? The SQLite amalgamation includes it by default.
Lunr.js isn't really being developed anymore from what I remember. I switched from Lunr.js to SQLite FTS in browser myself. I'm using sqlite-wasm-http [0], which uses range requests to only pull relevant pages of the static database as needed. Though if your search query is short enough, it'll probably pull the entire FTS table anyways.