Is there some way to silently, internally, redirect all pages to another server?

I have a site on Site5's server "huckleberry".
I can reach the page - with a SSL certificate - via the path to their server [this is common for shared hosts]:

1. <http://mysite.com/checkout.php> (insecure)
2. <https://huckleberry.site5.com/~myaccount/mysite.com/checkout.php> (secure)

All of the page's links and whatnot assume the first domain, and editing all of the pages is not practical.  

Eg. a link to /Assets/mystyles.css 404's out terribly.

Is there some way that users can log onto the second domain, and Apache will treat all requests as if they are pointing to the first page?

So that when the page requests /mystyles it will get mysite.com/mystyles.css instead of huckleberry.site5.com/~myaccount/styles.css?

I have tried to figure out using ProxyPass and mod\_rewrite, but can't get it to work.

Thanks