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>

src/components/ProtectedRoute.tsx

@@ -1,10 +1,25 @@
+import { useEffect } from 'react'
 import { Navigate, Outlet } from 'react-router-dom'
+import { authClient } from '../lib/auth-client.ts'
 import { useAuthStore } from '../stores/auth.ts'
 
 export function ProtectedRoute() {
-  const isAuthenticated = useAuthStore((s) => s.isAuthenticated)
+  const { data: session, isPending } = authClient.useSession()
+  const setAuthenticated = useAuthStore((s) => s.setAuthenticated)
 
-  if (!isAuthenticated) {
+  useEffect(() => {
+    setAuthenticated(!!session)
+  }, [session, setAuthenticated])
+
+  if (isPending) {
+    return (
+      <div className="flex min-h-screen items-center justify-center">
+        <div className="h-8 w-8 animate-spin rounded-full border-2 border-brand-blue border-t-transparent" />
+      </div>
+    )
+  }
+
+  if (!session) {
     return <Navigate to="/login" replace />
   }