rewrite HTML/JavaScript content from backend server

The correct way is to explain whoever wrote the website, the difference between relative and absolute URLs, and change those links to relative.

First result in google search:

http://www.dirigodev.com/blog/seo-web-best-practices/relative-vs-absolute-urls-seo/

If you don't want go that path, you can do that in F5, replacing the private FQDN with the public one.

You can use a stream profile, either adding to the virtual server, or an iRule to do some logic. Here are some solutions to explain how to use that:

https://support.f5.com/csp/article/K8115

https://support.f5.com/csp/article/K7027