fix(desktop): preserve selected model during session promotion

[usrnk1]

Issue for this PR

Closes#34466

Type of change

• Bug fix
• New feature
• Refactor / code improvement
• Documentation

What does this PR do?

Fixes a composer model picker flicker when submitting a new session with a non-default model selected.

During new-session promotion, the local draft model selection was cleared before the routed session state became available. That left the picker with no scoped selection for a moment, so it fell back to the configured default model, then switched back once the promoted session selection was restored.

This keeps the promoted model selection in an in-memory handoff state until the session route consumes it, so the picker continues showing the selected model throughout the transition.

How did you verify your code works?

• bun run typecheck from packages/app
• bun test src/components/prompt-input/submit.test.ts from packages/app

Screenshots / recordings

Before

Screenshot.2026-06-29.at.16.03.49.mp4

After

Screenshot.2026-06-29.at.19.06.55.mp4

Checklist

• I have tested my changes locally
• I have not included unrelated changes in this PR

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2. Add Fixes #<number> or Closes #<number> to this PR description

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[github-actions]

The following comment was made by an LLM, it may be inaccurate:

Potential duplicate found:

• PR fix(app): preserve draft model selection #33858: fix(app): preserve draft model selection
  fix(app): preserve draft model selection #33858

This PR appears to address the same issue—preserving the model selection in the composer/prompt input. The current PR (#34466) specifically targets the desktop context during session promotion, while #33858 may cover a related or overlapping scenario with draft model selection. You should check if #33858 already resolves this issue or if the fixes are complementary.