BulletProof Security is intentionally taking over your server's error page handling. Unfortunately they do this in a way that makes session handling impossible, breaking Akeeba Backup. There is a very simple workaround.
Create a new text file called .htaccess inside Akeeba Backup's folder, typically
that's wp-content/plugins/akeebabackupwp. Inside that file paste the following line:
ErrorDocument 403 /path-to-site/wp-content/plugins/akeebabackupwp/403.html
where path-to-site is the filesystem path to your site (e.g.
/home/cpaneluser/public_html) and akeebabackupwp is the folder
name where Akeeba Backup is installed. You can find the former by editing the .htaccess file in your site's
root. There is an ErrorDocument 403 line placed there by BulletProof security.
Finally, Create a new file called 403.html inside Akeeba Backup's folder (e.g.
wp-content/plugins/akeebabackupwp) with the following contents:
<html>
<head>
<title>403 Forbidden</title>
</head>
<body>
<h1>403 Forbidden</h1>
</body>
</html>
You only need to follow these instructions once, the first time you install Akeeba Backup or BulletProof Security.