feat: add WebVTT context generation to chat WebSocket endpoint

- Import topics_to_webvtt_named and recordings controller
- Add _get_is_multitrack helper function
- Generate WebVTT context on WebSocket connection
- Add get_context message type to retrieve WebVTT
- Maintain backward compatibility with echo for other messages
- Add test fixture and test for WebVTT context generation

Implements task fn-1.2: WebVTT context generation for transcript chat

.flow/specs/fn-1.8.md

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+# Task 8: End-to-End Testing
+
+**File:** N/A (testing)
+**Lines:** 0
+**Dependencies:** All tasks (1-7)
+
+## Objective
+Validate complete feature functionality.
+
+## Test Scenarios
+
+### 1. Basic Flow
+- [ ] Navigate to transcript page
+- [ ] Click floating button
+- [ ] Dialog opens with "Transcript Chat" header
+- [ ] Type "What was discussed?"
+- [ ] Press Enter
+- [ ] Streaming response appears token-by-token
+- [ ] Response completes with relevant content
+- [ ] Ask follow-up question
+- [ ] Conversation context maintained
+
+### 2. Edge Cases
+- [ ] Empty message (doesn't send)
+- [ ] Very long transcript (>15k chars truncated)
+- [ ] Network disconnect (graceful error)
+- [ ] Multiple rapid messages (queued correctly)
+- [ ] Close dialog mid-stream (conversation cleared)
+- [ ] Reopen dialog (fresh conversation)
+
+### 3. Auth
+- [ ] Works with logged-in user
+- [ ] Works with anonymous user
+- [ ] Private transcript blocked for wrong user
+
+### 4. UI/UX
+- [ ] Button doesn't cover other UI elements
+- [ ] Dialog scrolls properly
+- [ ] Streaming cursor visible
+- [ ] Input disabled during streaming
+- [ ] Messages clearly distinguished (user vs assistant)
+
+## Bugs to Watch
+- WebSocket connection leaks (check browser devtools)
+- Streaming text accumulation bugs
+- Race conditions on rapid messages
+- Memory leaks from conversation history