fix(flow-chat): resolve assistant mode from session workspace

[wsp1911]

Summary

Fix the AssistantScene mode regression where entering Assistant could temporarily treat the active project session as an assistant session and switch its mode to Claw.

Fixes #

Type and Areas

Type:

bug fix / regression fix / test

Areas:

web UI, flow chat

Motivation / Impact

ChatInput previously inferred assistant mode from the global current workspace. During the AssistantScene transition, the current workspace could switch to the assistant workspace before the input target session changed, which caused a normal project session to be misclassified and written back as Claw.

This change resolves assistant-mode detection from the target session's own workspace scope instead of the transient global workspace state, and adds regression coverage for the workspace-transition case.

Verification

• pnpm --dir src/web-ui run test:run src/flow_chat/utils/chatInputMode.test.ts
• pnpm run type-check:web

Reviewer Notes

The fix is intentionally scoped to ChatInput mode resolution. It does not change AssistantScene entry flow; it corrects the source of truth used when deciding whether the current input target should be pinned to assistant mode.

Checklist

• This PR is focused and does not include secrets, temporary prompts, generated scratch files, or unrelated artifacts.
• Relevant verification is recorded above, or skipped checks are explained.
• User-facing strings, docs, and locales are updated where applicable.