The solution was to increase the flexibility of our dashboard. I'm a huge fan of tiling window managers, so I rushed out to make one in JavaScript.
The result was window.js, a tiling manager for web sites. We hooked it up to our existing visualizations, did a bit of server-side magic to allow saving and loading, and voila-- 100% configurable, resizable, and sharable dashboards.
Check out the live demo here!