There was a time stopping DWA (DOORS Web Access) was necessary. Stopping DWA could be painful if it did not run as a service or that service was on another machine.
But that time has passed. Since 126.96.36.199 of DWA it will recover if it loses connectivity to the DOORS database. Ref: Interop Servers should tolerate DOORS servers being stopped and started (i.e. reconnect logic)
Rejoice. Then go simplify your backup scripts.