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/cartsnitch_api/auth/routes.py

@@ -1,4 +1,9 @@
-"""Auth routes: register, login, refresh, me, update, delete."""
+"""Auth routes: user profile management.
+
+Registration, login, refresh, and session management are handled by
+the Better-Auth service (auth/). This router provides user profile
+endpoints that query our own user data from the shared database.
+"""
 
 from uuid import UUID
 
@@ -8,10 +13,6 @@ from sqlalchemy.ext.asyncio import AsyncSession
 from cartsnitch_api.auth.dependencies import get_current_user
 from cartsnitch_api.database import get_db
 from cartsnitch_api.schemas import (
-    LoginRequest,
-    RefreshRequest,
-    RegisterRequest,
-    TokenResponse,
     UpdateUserRequest,
     UserResponse,
 )
@@ -20,37 +21,6 @@ from cartsnitch_api.services.auth import AuthService
 router = APIRouter(prefix="/auth", tags=["auth"])
 
 
-@router.post("/register", response_model=TokenResponse, status_code=status.HTTP_201_CREATED)
-async def register(body: RegisterRequest, db: AsyncSession = Depends(get_db)):
-    svc = AuthService(db)
-    try:
-        return await svc.register(body.email, body.password, body.display_name)
-    except ValueError as e:
-        raise HTTPException(status_code=status.HTTP_409_CONFLICT, detail=str(e)) from e
-
-
-@router.post("/login", response_model=TokenResponse)
-async def login(body: LoginRequest, db: AsyncSession = Depends(get_db)):
-    svc = AuthService(db)
-    try:
-        return await svc.login(body.email, body.password)
-    except ValueError:
-        raise HTTPException(
-            status_code=status.HTTP_401_UNAUTHORIZED, detail="Invalid email or password"
-        ) from None
-
-
-@router.post("/refresh", response_model=TokenResponse)
-async def refresh(body: RefreshRequest, db: AsyncSession = Depends(get_db)):
-    svc = AuthService(db)
-    try:
-        return await svc.refresh(body.refresh_token)
-    except ValueError:
-        raise HTTPException(
-            status_code=status.HTTP_401_UNAUTHORIZED, detail="Invalid refresh token"
-        ) from None
-
-
 @router.get("/me", response_model=UserResponse)
 async def get_me(
     user_id: UUID = Depends(get_current_user),