But, whose culture should be honored? Should the mountain have 2 different names, depending on whether you are approaching it from the Nepal side or the Tibet side?
The mountain needs one internationally recognized name. Otherwise, it would be too confusing for the mountain to have multiple names.
While Mt. Everest is not in Britain, the British did do most of the early exploration, surveying, scientific research, and mountaineering work in the Mount Everest region, so it is proper that the mountain has a British name. I understand also there are good reasons to use the local cultural name, but what happens when there are two or more cultures in the region? Which one do we use? In this case, it appears to me that calling it Mt. Everest is the proper solution.