test: cover public handler routes

[dominyyin-sudo]

/claim #71

Summary

• Add real Hono coverage for the public StarbaseDB handler routes.
• Verify /status/trace proxies the Cloudflare trace body and headers.
• Verify /api/* delegates to handleApiRequest.
• Cover JSON responses for unmatched routes and route-handler errors.

Tests

vitest run src/handler.public-routes.test.ts
✓ src/handler.public-routes.test.ts (4 tests)

vitest run src/handler.public-routes.test.ts src/handler.test.ts
✓ src/handler.test.ts (9 tests)
✓ src/handler.public-routes.test.ts (4 tests)

prettier --check src/handler.public-routes.test.ts
All matched files use Prettier code style!

I also ran the full suite locally. This new test file passed, and the only failures were the existing src/rls/index.test.ts expectations that already fail on current main:

Test Files  1 failed | 19 passed (20)
Tests       4 failed | 155 passed (159)