You might be better using the 'basic' http tests for downloads. You can do things like :
$response = $this->actingAs($user)->get(route('export.report'));
$response->assertStatus(200);
$response->assertHeader('content-disposition', 'attachment; filename="report.csv"');
$file = $response->getFile();
$filename = $file->getPathname();
// do whatever you need with the filename/file
@tobyreed when the ->get() method succeeds it implies the download was successful. If you also want to check the content of the downloaded file you could probably just get that from the $response.