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.