[router][grpc] Fix background tasks stored with wrong id

[CatherineSue]

Motivation

Background tasks were completing successfully but storing responses with the wrong ID:

• Background response ID: resp_b99907a8-77bd-4c94-8de5-3de9191a11a4
• Stored response ID: chatcmpl-1e093101-3e87-4913-a3de-0945e93402a5 (internal gRPC request ID)

This caused GET requests to return the original queued response instead of the completed one.

Root Cause:

The chat_to_responses() function was using chat_resp.id (the internal gRPC request ID) instead of the background response_id parameter.

Modifications

1. Created chat_to_responsesd() that accepts an optional response_id_override
2. Updated all call sites to use the new function and pass the response_id when available

Accuracy Tests

Before

Get response always shows the status is queued

After

Fixed. Able to get full response

Benchmarking and Profiling

Checklist

• Format your code according to the Format code with pre-commit.
• Add unit tests according to the Run and add unit tests.
• Update documentation according to Write documentations.
• Provide accuracy and speed benchmark results according to Test the accuracy and Benchmark the speed.

[gemini-code-assist]

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