I've gone through the TurboTax process and am at the very last step, which is to transmit my return for filing. However, when I click the button, it won't let me file.

This is most likely a browser issue.

Log out of TurboTax and Delete any Intuit-related cookies and empty the browser's Internet cache. Then close the browser, and reopen it. Try again, but on the sign-in page, be sure there is no check mark for "remember me" until you get this resolved.  

If still no luck, then do the above steps again, and if your browser has private browsing/incognito mode, try using a private window to access the TT site.

If still having problems, you can try a different browser if you have one available.