Laravel change language via URL/Route

Fairly new to Laravel and I'm trying to add a functionality that allows the user to switch between two languages by clicking a button in a header.blade.php file. So far I've got it so there's a test.php file in the respective lang directories with test strings and have managed to get <p>{{__('test.test')}}</p> to display the correct language when manually set. At the moment I'm not sure if this is actually calling the route to update the language or if the logic I have for updating it is wrong since I get no errors and I'm using barryvdh/laravel-debugbar to debug.

My logic for the button:

<button href="{{ url('language', config('app.locale') == 'en' ? 'fr' : 'en') }}">{{ config('app.locale') }}</button>

In routes/web.php:

Route::get('/language', 'LanguageController@show');
Route::post('/language/{lang}', 'LanguageController@update');

LanguageController.php, created via php artisan make:controller --api

<?php

namespace App\Http\Controllers;

use Illuminate\Http\Request;

class LanguageController extends Controller
{
    /**
     * Display the specified resource.
     *
     * @param  int  $id
     * @return \Illuminate\Http\Response
     */
    public function show($id)
    {
        return App::getLocale();
    }

    /**
     * Update the specified resource in storage.
     *
     * @param  \Illuminate\Http\Request  $request
     * @param  int  $id
     * @return \Illuminate\Http\Response
     */
    public function update(Request $request, $id)
    {
        //Tried the following
        config(['app.locale' => $id]);
        App::setlocale($id);
    }
}

Questions:

1. Is this the correct way to update the language at runtime?
2. How can I tell if my api calls are being made?
3. How can I achieve this inside of a template .vue file?
4. Is making a Controller for the language redundant?
5. Would the inner HTML of my button change if the locale was changed?
6. Is affecting config files at runtime bad practice?

--Edit-- I should also mention that the only reason I made a controller for this is because I had the route calls in web.php use a function instead however, they stated they were Closure running php artisan route:list and with the research I found I couldn't tell if that was correct

You are on the right way, but there is something missing.

You can't use the configuration to edit at runtime the language.

Save local language in user Session and create a new middleware to set on each request the language saved in session.

I found this article that can help you, localization-laravel