I was able to test Thanos hitting a BB-running minio server and add entries to the Thanos bucket
Essentially, we pass through any objstoreConfig configuration to Thanos and the PrometheusSpec. The objstoreConfig will vary widely depending on the actual object storage used. For a BigBig configuration, something like this will work:
In this instance, minio.minio.svc.cluster.local requires STRICT mtls. Because the monitoring.prometheus pod disables auto-istio mtls, we need to manually set up the tls certs here in order for the connection to work. These settings will vary depending on the environment, but the objstoreConfig should be flexible enough and passed straight-through to Thanos and the PrometheusSpec.