The cut-down SQL would include just the basics: CREATE TABLE, CREATE INDEX, INSERT, UPDATE, DELETE, SELECT. No triggers or views. No support for partial indexes or indexes on expressions, common table expressions, or other such features. There would be a minimal set of built-in SQL functions, but with support for application-defined SQL functions written in javascript.
A competing implementation would not need to understand the SQLite file format nor the complete SQLite language nor would it need to implement the SQLite API. All it would need to do is support the basic SQL subset defined by the spec.
If you get the browser vendors on-board with this idea, and I'll write the spec.