Not very sure but these features have potential to cause problem, because they may conflict with other plugins:
- Content Directory
- Database Prefix
- Filter Request Methods
- Filter Suspicious Query Strings
- Prevent long URL strings
Did you enable any feature above? Changing the Content Directory usually will cause problems with multisite environment, and the bad news are, you can't change it back it requires lots of work to repair once you enable the feature.
My suggestions (for mark_m):
1. Try to keep this Better WP Security activated but disable all features to see if your issue still happens.
2. Then try to enable the Hide Backend, and see if the issue will happen.
3. If you're using any caching plugin, you should only activate the caching plugin after you have make Better WP Security works correctly.
4. Try to use other theme to see if the issue still happens.