How can I test a multi-tenant Postgres app with in memory sqlite?

With our app, we're using Postgres as our DB. We have a public schema that houses all shared data, then each customer has their own schema (company_[ID]) with their own set of identical tables.

When a new customer comes on board, we create a new schema for them and migrate all tables to get them up to speed instantly.

When it comes to testing though, I'm struggling to find if there's some way we can speed things up by having the database be an in memory sqlite database.

Has anyone done anything similar, or have any suggestions?