[Omni] Refresh badge after deleting popup assets

[numbers-official]

Summary

This PR ensures the extension badge stays in sync after popup assets are deleted. It adds a background message path for refreshing badge state and has the popup trigger it after a successful delete, preventing stale badge counts or indicators.

Changes

• Add badge refresh handling in src/background/service-worker.ts for popup-initiated refresh requests.
• Update src/popup/popup.tsx to request a badge refresh after deleting popup assets.
• Extend shared message types in src/types/index.ts to include the new badge refresh message.

4 files changed, +28 additions, -2 deletions

File changes

.omni/e42f8a19-untitled/memory.md |  3 ++-
 src/background/service-worker.ts  | 11 +++++++++++
 src/popup/popup.tsx               | 13 +++++++++++++
 src/types/index.ts                |  3 ++-

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