Charlie, I don't think you are missing anything fundamental - it is an idiosyncracy. This is how Word chooses to apply Character styles - it removes local font settings when you apply it. You also can't stack Character styles (apply more than one at a time).

If you know your way around xml you could edit the document.xml to replace the Hyperlink style with MyStyle. This preserves the other local formatting.

When you need to do this - why do you need to do it? Could you modify the attributes of the Hyperlink style instead so that it looked like MyStyle?

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Andrew Lockton
Chrysalis Design, Melbourne Australia