feat(api): add managed Responses WebSocket sessions (#3887)

## Summary

Adds managed Responses WebSocket sessions with custom request headers,
independent lanes and final-response collection.

## Changes

- Preserve client headers and per-connection overrides during WebSocket
handshakes
- Route independent response lanes and collect final responses
- Reconnect explicitly with application restoration and no automatic
request replay

## Validation

Local WebSocket regressions cover custom request headers, independent
response lanes, connection lifecycle and final-response collection.
Current-head hosted checks pass. Authenticated synchronous and
asynchronous WebSocket tests verified custom headers and two-response
continuation.

Co-authored-by: markstuart-oai <323302876+markstuart-oai@users.noreply.github.com>