feat: update frontend path to '/badge' and adjust related documentation

frontend/src/main.tsx

@@ -6,12 +6,12 @@ import App from './App.tsx'
 import {DebugScreen} from './screens'
 
 function DebugWrapper() {
-    return <DebugScreen onBack={() => window.location.href = '/'}/>
+    return <DebugScreen onBack={() => window.location.href = '/badge'}/>
 }
 
 createRoot(document.getElementById('root')!).render(
     <StrictMode>
-        <BrowserRouter>
+        <BrowserRouter basename="/badge">
             <Routes>
                 <Route path="/" element={<App/>}/>
                 <Route path="/debug" element={<DebugWrapper/>}/>