Based on the OpenStreetMap example that you cited, I would say that it's possible but it could be quite a challenge. I would strongly advise against it since the OSM example uses version 2 of the Google API and it now officially deprecated.
But if you want to give it a try I would adapt the OSM example to point to the Bing tiles and make sure that the tileUrlTemplate
property matches Bing's format for storing tiles. Unfortunately, Bing uses a quad tree format while Google uses a coordinate based format for storing tiles and accessing them via a URL. It will be important to understand the differences if you're going to make the example work so be sure to dig into the documentation links above. Also, MapTiler has a fantastic visualization of the different tile formats out there. I've found this invaluable.
Personally, I would use OpenLayers. Since Bing and Google both use spherical mercator, adding multiple tile sources to a single map is a trivial exercise. Here is an example.