They've been using the two names for a long time. It's up there in the corner.
There may be a conflict between two kinds of national pride. One is wanting to go with the Hindi name and make the rest of the world use it and drop the foreign name. But "India" is known all around the world. "Bharat" doesn't have the same resonance. Burma could become Myanmar, because not that many people were familiar with the country. If India changes its name something would be lost.