I agree with everything the OP said above. Typically if you need to scale writes in SQLite, you'll want to look at sharding. The "single writer" restriction is per database so you can split your SaaS customers across multiple databases.

If your SaaS is in the hundreds or thousands of customers then you could split each customer into their own database. That also provides nice tenant isolation. If you have more customers than that you may want to look at something like a consistent hash to distribute customers across multiple databases.