I had a similar issue. I wanted to be able to catch the exception and redirect back with an error message. The trouble is that the exception is rendered before the Session starts.
I worked around this by editing App/Http/Kernel.php and moving the ValidatePostSize class from the middleware array into the middlewareGroups array -- directly after the StartSession class:
protected $middlewareGroups = [
'web' => [
\App\Http\Middleware\EncryptCookies::class,
\Illuminate\Cookie\Middleware\AddQueuedCookiesToResponse::class,
\Illuminate\Session\Middleware\StartSession::class,
\Illuminate\Foundation\Http\Middleware\ValidatePostSize::class,