Does not work for me WP 4.5.2 multisite

I have the same issue.

I have installed and activated version 1.1 of the Background Update Tester plugin under WordPress 4.6 multisite, but I cannot find the Updates → Update Tester link or the Dashboard → Update Tester screen.

Same in 4.7.x An no reply for 7 months. So dead as a project I guess.

Looks like it’s just the plugin’s menu hook that stopped working in multisite. Parent slug should be index.php, not update-core.php. Change line 27 of background-update-tester.php
from:
$top_level_menu = is_multisite() ? 'update-core.php' : 'index.php';
to:
$top_level_menu = 'index.php';

Here’s the diff if that’s easier for you, or if a maintainer wants to integrate it:

diff --git a/background-update-tester.php b/background-update-tester.php
index 0447ce0..4c222eb 100644
--- a/background-update-tester.php
+++ b/background-update-tester.php
@@ -21,7 +21,7 @@ class Background_Update_Tester_Plugin {

        function admin_menu() {
                load_plugin_textdomain( 'background-update-tester' );
-               $top_level_menu = is_multisite() ? 'update-core.php' : 'index.php';
+               $top_level_menu = 'index.php';
                add_submenu_page(
                        $top_level_menu,
                        __( 'Background Update Tester' , 'background-update-tester' ),