Wouldn't it be easier (if a little less slick-looking) to hang the date selection UI off an extension bar button rather than putting it into the actual web page? Then that permission could potentially be dropped.
Not sure what permissions are needed to get the current url from the plugin though.
But if it's really more about a good way to browse the way back machine I think a web-service/site would make more sense. Installing a browser extension is quite a hurdle for something you don't need that often.
Too bad bookmarklets are not really a thing anymore, that would work well.