Realtime API: Support 'base_model' cost tracking + show response in spend logs (if enabled) (#9897)

* refactor(litellm_logging.py): refactor realtime cost tracking to use common code as rest

Ensures basic features like base model just work

* feat(realtime/): support 'base_model' cost tracking on realtime api

Fixes issue where base model was not working on realtime

* fix: fix ruff linting error

* test: fix test

litellm/router.py

@@ -2138,6 +2138,7 @@ class Router:
                 request_kwargs=kwargs,
             )
 
+            self._update_kwargs_with_deployment(deployment=deployment, kwargs=kwargs)
             data = deployment["litellm_params"].copy()
             for k, v in self.default_litellm_params.items():
                 if (