feat: add fake payment API

[autochamchikim-pixel]

Summary

• Adds fake payment persistence to the in-memory database.
• Adds payment API routes for send, list, get, and status updates.
• Supports amount dollars or amount_cents and idempotency keys for retry-safe sends.
• Adds route tests covering send/list/get/update and duplicate idempotency behavior.

Test Plan

• npm exec --yes tsc --noEmit --pretty false
• npm exec --yes -- winterspec bundle --output dist/bundle.js
• bun test tests/routes/payments/send.test.ts could not be run in this container because Bun is not installed.

Closes #1