You already run infrastructure. A SaaS team hosting its own product can host its own support chat on the same footprint: Windows or Linux, behind IIS or Nginx, with SQL Server as the store.
Your users' data is a trust asset. Keeping support conversations in your own database, rather than a third-party messenger, keeps a sensitive channel inside the boundary you already secure and audit.
Costs stay flat as you scale. Add agents and handle more conversations without a per-seat meter. The AI provider is yours to choose, so inference cost is under your control too.