feat: migrate authentication to Better-Auth (Phase 1)

Replace hand-rolled JWT auth with Better-Auth session-based authentication.

- Scaffold auth/ Node.js service with Better-Auth, bcrypt password compat,
  Postgres adapter mapped to existing users table
- Add Alembic migration (002) creating sessions, accounts, verifications
  tables and migrating password hashes to accounts table
- Update FastAPI auth dependency to validate sessions via shared DB
  (supports both cookie and Bearer token)
- Remove registration/login/refresh endpoints from API gateway (now
  handled by Better-Auth service)
- Update frontend to use better-auth/react client with httpOnly cookies
  (no tokens in localStorage or memory)
- Rewrite auth store, Login, Register, Dashboard, Settings, ProtectedRoute
  to use session-based auth
- Update all tests to create sessions directly in DB instead of JWT tokens

Resolves CAR-27
See plan: CAR-26#document-plan

Co-Authored-By: Paperclip <noreply@paperclip.ing>

tsconfig.json

@@ -0,0 +1,16 @@
+{
+  "compilerOptions": {
+    "target": "ES2022",
+    "module": "ESNext",
+    "moduleResolution": "bundler",
+    "outDir": "dist",
+    "rootDir": "src",
+    "strict": true,
+    "esModuleInterop": true,
+    "skipLibCheck": true,
+    "declaration": true,
+    "resolveJsonModule": true
+  },
+  "include": ["src"],
+  "exclude": ["node_modules", "dist"]
+}