I've found that the best solution is to _not_ make your website "just another client" of your API. On the initial page load, render the page normally using your server-side templating. Then have your JavaScript introspect the existing DOM and enhance it into a very interactive close to single-page app, using client-side rendering from that point forward. Yes, there's code duplication with templates, and you have to deal with certain peculiarities like implementing a user event queue between the time the page loads and the JavaScript is done hooking into everything so you don't lose user interaction. But those are pains I'm willing to deal with.
That's a similar technique to what I'm doing; I use language-agnostic templates (mustache) so that I can do a full server side render, and then my Backbone app initializes with bootstrapped data. This way, we can progressively enhance the site and have a quick page load and at the same time build a single page Backbone app.
I've been using hogan.js to compile mustache templates to JavaScript functions on the server-side and write them to a js file that is then served to the browser:
