Since the maintenance page is a feature of the web application, there is no reason to put it into a separate branch or repository. Instead, have a dedicated directory, or maybe even just a file at the root. This will make it possible to:
Share all static files such as the images, CSS or JavaScript,
Benefit from common code instead of duplicating it with the risk of the maintenance page being not properly updated when the web application itself changes.
You probably won't be able to share everything, especially since, I presume, your web app uses templates while the maintenance page is static, but it's still much less hassle than to maintain a different branch or repository.
You can then switch between the web application and the maintenance page by changing the configuration of the server (like an automatic redirection of any GET request other than favicon.ico and robots.txt to the maintenance page, and an HTTP 500 for POST requests).