feat: serve the app under a base path

[adamdickinson]

Serve the app under a base path

Estimated time to review: 10 min (Small)

🎯 Intent
Lets the app run behind a proxy that serves it under a path, without changing how it runs anywhere else. With BASE_PATH set, the app, its links and the mock API all live under that path. Without it, everything stays at the root, as today.
⚠️ Risks	🔍 Focus
⚠️ Watch: the mock API middleware runs ahead of Vite's own, so it strips the base itself before handing the request to handlers written against the root.	Look at: vite-plugin-mock-api.ts Skim: vite.config.ts, src/main.tsx, src/lib/api-client.ts, which each read the base from Vite

Motivation

The app assumed it was served from /: the router's links, the API client and
the mock API middleware all used root paths. Behind a proxy that forwards a path
such as /preview/, the page loaded but its routes and API calls missed.

Verification

1. Run BASE_PATH=/preview/ bun run dev and open http://localhost:5173/preview/. Confirm the app loads, its scripts load from under /preview/, and /preview/api/me answers with JSON while /api/me is a 404.
2. Run bun run dev with no BASE_PATH and confirm the app and /api/me still answer at the root.
3. Run bun run typecheck, bun run lint and bun run test, and confirm they pass unchanged.

🤖 Generated with Claude Code