fix: resolve failing tests and type errors across api and shared packages

[stooit]

Summary

Fixes all failing tests and type errors in the multi-package TypeScript HTTP API (Hono). After this change: 22/22 tests pass and tsc --noEmit is clean.

Bugs fixed

• Auth middleware (packages/api/src/middleware/auth.ts) — case-sensitivity bug in the public-route allow-list. Hono normalises c.req.method to uppercase, so the lowercase comparison never matched, causing public routes (e.g. POST /users) to wrongly require a token. Now compares against uppercase method names.
• Shared types (packages/shared/src/types.ts) — field-name inconsistency between packages. The User type field is now consistently username, matching what the API routes and tests expect.
• Users route (packages/api/src/routes/users.ts) — missing badRequest import caused a runtime failure on the 400 (missing fields) path.
• Pagination utility (packages/shared/src/utils/pagination.ts) — implemented the paginate stub to satisfy the full test contract (page slice, partial last page, out-of-range page, total/totalPages, page/pageSize).
• tsconfig — added bun-types (already installed, no new dependency) so bun:test and the process global resolve under tsc.

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → exit 0, no errors

Notes / assumptions

• Constraint honoured: no test files modified, no new dependencies added — only the source changes the tests required.
• Field-name inconsistency was resolved by aligning source to the tests' expectation (username), since tests must not be edited.
• A code review flagged two latent edge cases not covered by tests (size=0 and negative page in paginate). These were left untouched to respect the "fix only what the tests require" constraint and avoid scope creep.

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