Hi Chris,
We have imported setting options from your website and found we have an option to open menu on page load that is “Show Menu on Page Load” under advance tab.
And also tested this on my local and working fine but in customer site have some different issue as well that is conflict with “Float Menu Pro”.
You are using two menu plugins together and may be those are conflicting.
As this is your live site so we can not directly change anything on this. Please off this “Show Menu on Page Load” and remove the cache if it is working fine otherwise.
Remove all other plugin and just keep responsive-menu-pro with wordpress default theme. And after that still have an issue then please give us the ftp access for debugging.
Regards,
Kriti