I received your BPS Pro Log files and this is the problem that occurred:
What it looks like occurred was that your host was trying to update WordPress on your website starting on May 16 when WP 4.7.5 was first available. It appears your host kept trying to force upgrade WordPress on your site to WP 4.7.5 continuously for several days at different intervals.
BPS Pro AutoRestore works automatically for WP Automatic Updates and Manual WordPress Updates (Dashboard > Updates > Install Now) and will automatically backup any new WordPress files that have been installed/updated.
BPS Pro AutoRestore will not work correctly when a web host force upgrades WordPress on a website because that exactly simulates a hacker changing/adding files to your website. AutoRestore is amazing, but it cannot tell who is changing/adding files to your website if that is being done externally and not from within WordPress itself by you or by the WordPress API server.
So what you want to do to prevent this problem from occurring in the future is ask your web host to disable WordPress forced upgrades on your website and then enable WordPress Automatic Updates on your website so that WordPress itself will install/update your website automatically.