Firstly, your website is impossible to navigate on iPhone - hugely large and doesn't shrink to fit the viewport. It also keeps crashing when I try to change your switcher. This is another matter entirely but it means that I can only really make suggestions for your queried problem.
I imagine the frames are what is causing the 'half-switch' - the stylesheet immediately changes in the document where the select options are located, but until there is a refresh, the cookie cannot apply the change to the other framed documents that make up your site 'page'. A workaround for this could be to add a refresh option into the select menu for the style switcher - to force a page refresh on theme selection.
A reload could be as simple as this;
I haven't tested but here are two examples for reference;