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>

package.json

@@ -0,0 +1,24 @@
+{
+  "name": "@cartsnitch/auth",
+  "version": "0.1.0",
+  "private": true,
+  "type": "module",
+  "scripts": {
+    "dev": "tsx watch src/index.ts",
+    "build": "tsc",
+    "start": "node dist/index.js",
+    "generate": "npx @better-auth/cli generate"
+  },
+  "dependencies": {
+    "better-auth": "^1.2.0",
+    "pg": "^8.13.0",
+    "bcrypt": "^5.1.1"
+  },
+  "devDependencies": {
+    "@types/node": "^22.0.0",
+    "@types/pg": "^8.11.0",
+    "@types/bcrypt": "^5.0.2",
+    "tsx": "^4.19.0",
+    "typescript": "^5.7.0"
+  }
+}