Conroy Whitney bbf2205640 feat(gateway): inject timestamps into chat.send (webchat/TUI)
The chat.send handler (used by webchat and TUI) is a separate path
from the agent handler. Inject timestamp into BodyForAgent (what the
model sees) while keeping Body raw for UI display.

This completes timestamp coverage for all non-channel paths:
- agent handler: spawned subagents, sessions_send, heartbeats
- chat.send: webchat, TUI
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