Reuse bridge-owned Claude writer for repeated MCP turns

[Yehonal]

Summary

• Reuse a Claude CLI writer only when this gateway process resumed it and the writer PID and single unattached tmux pane still match.
• Keep external CLI and Desktop-owned sessions on their existing fail-closed or managed inbox paths.
• Cover two consecutive turns and writer/pane replacement in tests.

Validation

• npm run verify passed on the branch projected from current main.
• git diff --check origin/main...pr/claude-mcp-owned-writer passed.

Rollout

• Gateway rollout is required before live behavior changes. Ownership proof is process-local and does not adopt a writer surviving a gateway restart.
• Verify with two harmless correlated MCP turns in the same persisted Claude session; inspect the actual transcript responses, not simulated delivery.