How to dynamically add "integrationInstance" parameter to VBCS API calls to OIC Gen3?

I have a VBCS application that connects to the factory OIC API for administrative tasks that the OIC admin UI doesn't handle well. This application was initially written for OIC Gen2, and we've now migrated to Gen3, which has some differences.

I defined a custom backend with Dev, Test, and Prod servers, all using OAuth2 client_credentials. Each environment has its own Confidential Application in IDCS with separate credentials. In Gen2, the base URL varied because the instance name was embedded in the subdomain. In Gen3, the base URL is the same for all environments, and we need to provide a dynamic querystring parameter called "integrationInstance" to direct the API call to the correct Gen3 instance.