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Without knowing exactly how styleswitcher.js works, it's hard to give you a solid answer.
Is this a style choice that your USER makes on the main page? Or does the code check for BROWSER version and automatically switch stylesheets as needed?
It does seem to me that you could pass a variable to the iframed document, or have a script in the head of that document that looks at the variable in the parent window. If the iframe is alread loaded, the styleswitcher.js would also need to force a reload for the iframe.
The specifics might get a bit messy, but it sounds do-able.