Remote Control (/rc) makes local interactive session noticeably slower

## Summary
With Remote Control connected via `/rc`, the local interactive terminal session feels noticeably slower to respond. Immediately after disconnecting with `/remote-control`, responses become fast again.

## Steps to reproduce
1. Start `claude` (interactive)
2. Run `/rc` to connect Remote Control
3. Chat locally in the terminal — responses are noticeably delayed
4. Run `/remote-control` to disconnect (`Remote Control disconnected.`)
5. Responses are fast again

## Expected
No perceptible latency penalty on the local session while Remote Control is connected.

## Environment
- Claude Code 2.1.270
- macOS (Darwin 25.6.0), zsh
- `remoteControlAtStartup` not set (RC was enabled manually with `/rc`)

## Notes
Docs say the WebSocket transport keeps latency minimal, but the local slowdown was consistent across the session and disappeared the moment RC was disconnected.

No source file, test, or entry point is named. Reproduce the slowdown in an interactive Claude Code terminal by connecting with /rc, then disconnect with /remote-control and compare responsiveness; done means no perceptible local latency penalty while Remote Control remains connected.

Written by the indexing model from the issue text.