How about having two pools, one for writes only, and the other one for reads? SQLite allows you to open the DB more than in one thread per application, so you can have a read pool and a write pool with SetMaxConnections(1) for better performance. This of course also means that reads should be handled separately from writes in the API layer too.
Well I agree, that's a good starting point. You probably won't be able to beat Redis with SQLite anyway :), although given that WAL mode allows for concurrent reads it might give it a large enough performance boost to match Redis in terms of QPS if the concurrency is high enough.