The REST classes automatically unzip the file for you (and it also unchunks it for you). The resultant value is given to you in a binary format that is already a PDF. Therefore, using renderAs="PDF" will attempt to render the PDF as a PDF using the native HTML-to-PDF translator, which as you might now realize, won't work.

What you need is to specify a contentType on the page of "application/pdf", do not use a renderAs attribute, and somehow be able to output the binary string to the page. This is the primary sticking point that several developers have experienced in the past (and of which I sadly have no "straight" answer to)-- you can't choose to output binary data in any meaningful fashion in Visualforce.

There's simply no way to output a pre-rendered anything (zip file, pdf file, image, etc) to the output stream in a binary format, or by specifying a content transfer encoding that the browser would accept and decode correctly. The only "legal" way in the platform to even output raw binary data today is Base64 encoding, which would work if you could specify a Content-Transfer-Encoding header, which we can't (not that I've seen).

The short-term solution that may work, is as follows:

1) Retrieve your content from the server.

2) Save the file to a Document or an Attachment.

3) Return a PageReference to the Document or Attachment download link.

I've never actually tried this so I'm not 100% confident it would work this way; I don't have the time to write a full-scale mockup required to test this. The actual limitation would come down to if the REST classes can provide a binary response back (which I don't think they can at the moment). If your server can pre-encode to base64, then the response text would be binary-encoded, which you could then directly save into the aforementioned document or attachment; the EncodingUtil class would help you out here, as you could decode the base64 back into a native Blob format.