## Bug Description

Router deployments can set per-deployment `num_retries` and `cooldown_time` in `litellm_params`, but those settings are not consistently honored for the generic Router paths used by:

- `router.aresponses(...)` - `router.aanthropic_messages(...)`

The chat/completions Router path already has plumbing that stamps the selected deployment's retry settings onto raised exceptions before the Router retry loop handles them. The generic Responses / Anthropic Messages path selects a deployment and forwards the request, but when the provider call raises, the selected deployment's `num_retries` is not propagated to the retry loop.

Separately, the failure callback chooses cooldown duration from `kwargs["litellm_params"]["cooldown_time"]` when present, falling back to the retry header or Router-wide `cooldown_time`. The Responses and Anthropic Messages logging params include `model_info`, but do not currently include the selected deployment's `cooldown_time`, so deployment-level cooldown overrides are skipped on those paths.

## Expected Behavior

For routed Responses and Anthropic Messages calls:

1. A deployment-level `litellm_params.num_retries` should override the Route…