How to handle a fixed element when the viewport is narrower than the page content (resulting in horizontal scrolling)?

Our designers and product managers like to use fixed elements in some pages of our site to keep something persistent. The problem is when a user makes their browser narrower than our page (which has a pixel-defined width), resulting in horizontal scrolling, when he/she scrolls left to right and back, the fixed element stays put while the rest of the content scrolls. Another thing to note is that it's also a problem because per the mocks, the fixed element is supposed to look like it's a part of the main content i.e. they expect the fixed element to visually "stay" with the content. But as I pointed out, this doesn't happen when the viewport is narrower than the page. The fact that it stays put while the rest of the content moves is expected behavior with fixed elements, but of course they don't like that. Here's a pen of a simplified version of what I'm talking about - http://codepen.io/annmarriott/pen/wWpvzO. Narrow your browser to see the effect.

What I've seen on other sites is that when horizontal scrolling is triggered, these sites will simply hide the fixed element. This is not acceptable by the good folks I work with. I think there are JavaScript solutions, but from what I've seen they are not great. They result in janky motions. I'm not a JS expert, but it seems to me that JavaScript would have to constantly and dynamically calculate the CSS left property value to be applied to the fixed element as the user is scrolling back and forth. That would result in a visually jarring experience.

What would you do in the horizontal scrolling scenario? Hide the element? Use a JavaScript solution to keep it moving with the content? Come up with something better than a fixed element? Something else? Personally I would do nothing. Most of our users will not narrow the browser window, and if they do, they won't be navigating the site in that width. The site is very hard to use when the window is that small. Also we don't have the option of changing the pixel-based width. Our web app has been around for a long long time, so changing that would require a HUGE refactor taking several weeks. We have millions of users, and it's a big risky undertaking for a couple of fixed elements.