You would need to move the code from one template to another. That means you will need to preregister the variable for use in the new template. Cellarius wrote a really good article that you may be interested in to help with that -
[vB4] Rendering templates and registering variables - a short guide Also, you'll need to make sure all the php code to create that dropdown is rendered on every page your want to have the dropdown on and that may mean you need to copy it to a plugin.