[Remote] Assistant sessions display the wrong workspace and branch

## Bug
When opening a Claw assistant session in BitFun Remote, the header displays `home / master`.

## Expected behavior
Assistant/Claw sessions should show the assistant context, or omit repository workspace and branch information when no Git workspace is attached.

## Actual behavior
The remote web UI keeps displaying the previously selected normal workspace (`home`) and its branch (`master`). This is misleading, although command execution appears to use the correct assistant workspace.

## Reproduction
1. Open BitFun Remote with a normal workspace selected.
2. Switch to Assistant mode.
3. Open a Claw assistant session.
4. Observe the session header: it still shows the normal workspace and branch.

## Environment
- BitFun 0.2.17
- macOS desktop client
- BitFun Remote web UI via relay

## Additional evidence
The Claw session target is the dedicated assistant workspace (`~/.bitfun/personal_assistant/workspace`), which is not a Git repository. The desktop runtime logs also report that assistant session commands use that path, so this appears to be stale web UI state rather than backend routing to the normal workspace.

Reproduce this in the BitFun Remote web UI by selecting a normal workspace, switching to Assistant mode, and opening a Claw session. Trace how the session header retains workspace and branch state; done means the assistant context is shown or repository fields are omitted, without changing normal workspace behavior.

Written by the indexing model from the issue text.