Hi there
I haven't played a lot with frontend stuff for a while, so my answer might be outdated, but it should work.
I used to use Charles Proxy (because it's so useful for so many debugging stuff), which has a nice feature: rewrite. (Warning, the steps I mention here are from an old version, but my license isn't valid for the latest one so I stick with this, I'm confident it still offer this feature, but the menu options might have moved or been renamed)
In the Tool -> rewrite... menu , you can enable rewrite then add a set of rules. In that set you create, you had the location in which this should apply (www.website.com in your case, to only apply this rewrite set of rules to that host), and you had a rule, like Add-Header, (or modify header, depending exactly of what is missing), and boum, you added cors headers and your browser should be happy!
Charles works on Mac and Windows. If you're on Windows, then Fiddler is free and much more powerful and let you do the same kind of rewrite (but I can't guide you there, I'm on osx and don't have such a good memory to remember all the menu names ;-)