If you want to allow the user to use whatever PDF viewer they have configured, you cannot add functions to that viewer. If you want to control the operation of the viewer, you need to provide it yourself. This is very much an either/or situation. Attempting to mix the two will not lead to good results. Google and Mozilla are free to change their PDF viewers without giving any thought whatsoever to your site.
As mentioned above, you can embed PDF.js in your web site, per the instructions at https://github.com/mozilla/pdf.js/wiki/Setup-pdf.js-in-a-website. I don't know whether this viewer's PDF support is good enough for your scenario, but you can certainly test it out. Since the PDF.js code is hosted on your site and thus under your control, you may be able to edit it as you see fit.