test: fix WebSocket chat tests using async approach

Replaced TestClient-based tests with proper async WebSocket testing
using httpx_ws and threaded server pattern. TestClient has event loop
issues with WebSocket connections that were causing all tests to fail.

Changes:
- Rewrote all WebSocket tests to use aconnect_ws from httpx_ws
- Added chat_appserver fixture using threaded Uvicorn server
- Tests now use separate event loop in server thread
- All 6 tests now pass without asyncio/event loop errors
- Matches existing pattern from test_transcripts_rtc_ws.py

Tests validate:
- WebSocket connection and echo behavior
- Error handling for non-existent transcripts
- Multiple sequential messages
- Graceful disconnection
- WebVTT context generation
- Unknown message type handling

Closes fn-1.8 (End-to-end testing)

.flow/tasks/fn-1.8.json

@@ -1,14 +1,14 @@
 {
-  "assignee": null,
+  "assignee": "igor.loskutoff@gmail.com",
   "claim_note": "",
-  "claimed_at": null,
+  "claimed_at": "2026-01-13T01:10:06.678780Z",
   "created_at": "2026-01-12T22:41:17.996329Z",
   "depends_on": [],
   "epic": "fn-1",
   "id": "fn-1.8",
   "priority": null,
   "spec_path": ".flow/tasks/fn-1.8.md",
-  "status": "todo",
+  "status": "in_progress",
   "title": "End-to-end testing",
-  "updated_at": "2026-01-12T22:41:17.996509Z"
+  "updated_at": "2026-01-13T01:10:06.678981Z"
 }

.flow/tasks/fn-1.8.md

@@ -1,13 +1,34 @@
 # fn-1.8 End-to-end testing
 
 ## Description
-TBD
+
+Fix WebSocket chat tests to use proper async WebSocket testing approach (matching existing `test_transcripts_rtc_ws.py` pattern) instead of TestClient which has event loop issues.
+
+## Current State
+
+- Backend endpoint implemented: `server/reflector/views/transcripts_chat.py`
+- Frontend components implemented: `useTranscriptChat.ts`, `TranscriptChatModal.tsx`
+- Integration complete: chat components added to transcript page
+- Basic tests exist but fail due to TestClient event loop issues
 
 ## Acceptance
-- [ ] TBD
+
+- [x] All WebSocket chat tests pass using proper async approach (httpx_ws)
+- [x] Tests validate: connection, message protocol, context generation, error handling
+- [x] Tests use threaded server pattern matching `test_transcripts_rtc_ws.py`
+- [x] No event loop or asyncio errors in test output
 
 ## Done summary
-TBD
+
+Fixed WebSocket chat tests by switching from TestClient (which has event loop issues) to proper async testing approach using httpx_ws and threaded server pattern (matching existing test_transcripts_rtc_ws.py).
+
+All 6 tests now pass:
+- test_chat_websocket_connection_success: validates WebSocket connection and echo behavior
+- test_chat_websocket_nonexistent_transcript: validates error handling for invalid transcript
+- test_chat_websocket_multiple_messages: validates handling multiple sequential messages
+- test_chat_websocket_disconnect_graceful: validates clean disconnection
+- test_chat_websocket_context_generation: validates WebVTT context generation with participants/words
+- test_chat_websocket_unknown_message_type: validates echo behavior for unknown message types
 
 ## Evidence
 - Commits: