[Bug]: Responses API mid-stream 429 can be recorded against the fallback deployment and cool it down

Finding: In an external report on LiteLLM proxy v1.97.0 (https://github.com/BerriAI/litellm/issues/45645), a streaming /v1/responses call that ends with an in-stream rate_limit_exceeded error can lead the router to cool down a healthy fallback deployment instead of the primary that returned the 429. Once every deployment in the fallback group has been cooled down this way, direct requests to that group fail with a no-deployments-available error (cooldown_list=['backup-a', 'backup-b'] in the report). Sources: https://github.com/BerriAI/litellm/issues/45645. Reported conditions: the primary model group has one deployment and falls back to a group with two deployments, with cooldown_time 60 and a sync failure callback. The primary's stream emits an error event with code rate_limit_exceeded, which LiteLLM turns into MidStreamFallbackError with status 429 and which starts the mid-stream fallback. The reporter's fake fallback backend returns HTTP 400 after 0.5 s, and the fallback deployments were never sent a 429 (https://github.com/BerriAI/litellm/issues/45645). Timing dependence: the reporter says a deliberate 150 ms sleep in a sync failure callback made it happen on every request. Without it, the proxy showed 0 of 60 hits on a laptop. Router-level loops showed 14 and 18 of 30 on 1.97.0 and 6 of 30 on 1.104.0 source. The 1.104.x proxy was not run (https://github.com/BerriAI/litellm/issues/45645). Reporter's hypothesis, not maintainer-confirmed: the Responses streaming iterator waits for the async failure handler but submits the sync failure_handler to a thread pool without waiting. The router's cooldown logic lives in the sync handler. Meanwhile the fallback attempt updates the shared logging object with the fallback deployment's model_info, so when the delayed handler runs, Router.deployment_callback_on_failure reads the fallback deployment's id and cools that one down (https://github.com/BerriAI/litellm/issues/45645). Version scope: the reporter says this path is unchanged in substance in 1.104.0, 1.104.1, 1.104.2 and main at 18e87d3039, based on source reading plus a Router-level test on 1.104.0 (https://github.com/BerriAI/litellm/issues/45645). Status: no maintainer response and no confirmed fix or release are in the supplied data. The reporter suggests a per-attempt copy of the logging object, as in open PRs #45136 and #45194 for /v1/messages, or carrying the failing deployment id on the exception. These are proposals only. #45194 reportedly does not touch the Responses streaming iterator. Limitation: this is an external report only; KFA did not reproduce it. No workaround is supported by the source. For diagnosis, the reporter's signature is a cooldown log entry for a fallback deployment carrying a MidStreamFallbackError 429 while the fallback upstream received no 429.