That’s not an issue to fix. Based on the theme UX the Client Area Button will not show in the small device (mobile). Because it destroys the header. You could check our theme live demo as a reference, where also not showing the button on mobile view.
We have reviewed your WHMCS and saw you are using template six and the issue also exists in the template six. That means this is not our theme issue. It’s all about WHMCS issue. In this issue, please contact with WHMCS support to get a solution.