I'm not sure what's complicated about this. Instead of exposing a database instance as the write master, you expose a service as the write master. You can even do it transparently, if your load balancing fabric cooperates: just arrange to have read replicas redirect write requests to the write master.
With that setup you have essentially lost any benefits that SQLite provides, specifically any performance benefits by making a DB read/write a network call instead of a local call in the same process that manipulates some file on the same machine. You are closer to a postgres/mysql patterne except now your writes don't scale as well. What are we actually gaining with this setup other than perhaps easier administration of the "db service" compared to a mysql/postgres deployment?
Obviously, you have not lost those benefits: reads are satisfied from an in-process database, and, for most applications, most requests are reads.