You can delete the current cookie (potentially cookies plural) and other critical information (the aliId and mkt_tok query params) without reloading the page, and you can generate a new cookie as well. The browser's cookie store is always updated synchronously.
But the reason the Not You? link reloads the page is because you can't uninitialize Munchkin after the page loads and Munchkin has already logged the initial Visits Web Page activity and attached related click event listeners.
The assumption is if you have a bad session association, you want Munchkin to start over with a new Visit Web Page (and subsequent Click Link events) for an anonymous lead.
If you don't mind the fact that the session will still be associated with the old lead until the person navigates away, you can reset the cookie without reloading, then re-render a new form with the new cookie. I wouldn't try to get this working unless you understand all the moving parts thoroughly, though, and your Not You? should only be clicked a handful of times per day at best -- it's not generally part of people's UX.
You can read all about the Not You? link at