How to transfer a panel variant from one site to another? Is there a way to do this with Features? I can't download and install any additional plugins to do this on this site. 

Would anyone know if this is possible?
Yes, Panels and panel variants can be exported using the Features module. If you are using the panel variant as a part of page manager the option to export variant is shown in the features UI under page_manager_handlers.

Check the screenshot below to see how sample variants of the node view (node/%node) page can be exported via features: