[FEATURE] Add `/fork` and `/branch` (conversation branching) support to Remote Control web/mobile clients

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

The CLI supports `/branch` (clone the conversation, switch into the new copy, leave the original untouched) and `/fork ` (spawn a background subagent that inherits the conversation and works independently). This has already been requested for other surfaces that lack it: the VS Code extension (#69272), the JetBrains plugin (#93301), and the Cursor extension (#93379).

Remote Control has the same gap, for both commands:

- Per the [Remote Control docs](https://code.claude.com/docs/en/remote-control), under **Limitations > "Some commands are local-only,"** neither `/fork` nor `/branch` is in the list of commands supported from mobile/web during a Remote Control session (only `/compact`, `/clear`, `/context`, `/usage`, `/exit`, `/usage-credits`, `/recap`, `/reload-plugins`, `/model`, `/effort`, `/fast`, `/color`, `/rename`, `/mcp`, `/config`, `/autocompact`, and `/advisor` are listed as working remotely).
- Attempting `/fork` from claude.ai/code or the Claude mobile app while connected via Remote Control fails at runtime with a generic error, "Couldn't fork this session," instead of being disabled or explained.
- `/branch` doesn't even produce an error — it's simply not recognized as a command at all in the web view, with no feedback of any kind.

### Proposed Solution

- Add `/fork` and `/branch` support to Remote Control's web and mobile clients, matching their CLI behavior.
- At minimum, if full support isn't feasible soon, both commands should fail with a clear message stating they're local-only (consistent with how `/resume` and `/plugin` are documented), rather than `/fork`'s generic runtime error or `/branch`'s silent non-recognition.

### Alternative Solutions

For `claude remote-control` (server mode), there is no workaround at all: the local terminal only displays connection status and logs, with no input prompt to type `/fork` or `/branch` into locally. The only way to reach either command is to stop server mode and start an interactive session instead (`claude --remote-control`), which does have a local prompt — but that changes the whole session model and isn't a real substitute for server mode's multi-session design.

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

1. I'm steering a session on my home machine via claude.ai/code from my phone, with `claude remote-control` running in server mode.
2. Claude proposes an approach mid-session and I want to try a different direction without losing the current one.
3. On the CLI I'd `/branch` to get a separate session I can go back and forth between, or `/fork ` to delegate a variation as a background task.
4. From Remote Control's web view, I can't do either: `/fork` fails with "Couldn't fork this session," and `/branch` isn't recognized as a command at all. There's also no local terminal prompt to fall back to, since server mode only shows logs.

### Additional Context

- Same underlying request as #69272 (VS Code), #93301 (JetBrains), #93379 (Cursor) — this covers the Remote Control web/mobile surface specifically.
- Environment: `claude remote-control` (server mode) on a self-hosted machine, attempting `/fork` and `/branch` from claude.ai/code (web view).

Start with the Remote Control documentation's Limitations section and reproduce the reported `/fork` and `/branch` behavior from a web or mobile session. Compare the remote behavior with the CLI commands; done means both commands work remotely with matching behavior, or consistently report that they are local-only instead of failing generically or being silently unrecognized.

Written by the indexing model from the issue text.