Very welcome. I think this is going to turn out to be just one of those wierd cPanel issues. All the clues are indicating some kind of cPanel setting issue, but there are other strange sorts of problems that occur randomly like directory corruption/permission corruption or 0 byte directory issues and some other very difficult to troubleshoot issues that are not obvious. So yep let me know what the cause and solution are once you/your Host get this sorted out. Thanks.
All BPS Problems are solved by deleting the BPS plugin and deleting the root and wp-admin .htaccess files so when those things are done and the problem is still occurring then BPS was not being naughty in that case. ha ha ha. ??