### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

In a multi-replica LiteLLM proxy deployment with `usage-based-routing-v2`, the deployment-level TPM limit (`litellm_params.tpm` in `model_list`) is enforced against each replica's local in-memory counter rather than the cross-pod sum. The effective per-deployment TPM ceiling becomes `tpm_limit × N_replica`, and traffic up to that ceiling passes through with zero 429 responses.

### Root cause (code-level)

**RPM keys are batch-synced across replicas** (since #9357 — "support batch writing increments to redis"), but **TPM keys are not**:

`lowest_tpm_rpm_v2.async_log_success_event` (line ~305): ```python await self.router_cache.async_increment_cache( key=tpm_key, value=total_tokens, ttl=..., parent_otel_span=..., ) ```

This calls `DualCache.async_increment_cache`, which writes to both in-memory and Redis directly, **but does not register the key with the queue-based sync mechanism** in `base_routing_strategy.py`. The periodic `_sync_in_memory_spend_with_redis` task only processes keys that go through `_increment_value_in_current_window` (wh…