Both names are official and interchangeable and are in different languages -- Hindi/Sanskrit and English.
"Bharat" and "India" have always been used officially for our Country in most of the official names and documents.
Bhartiya Prodyogiki Sansthan, Indian Institute of Technology
Bhartiya Nausena, Indian Navy
Bhartiya Rail, Indian Railways
Bharat Sanchar Nigam Limited
Bhartiya Reserve Bank, Reserve Bank of India
Akhil Bhartiya Ayurvigyan Sansthan, All India Institute of Medical Sciences
Bhartiya Prashasnik Seva, Indian Administrative Services
My father used to work in DoT/BSNL in long distance networks. He used to go on petroling the long distance cable networks. All his vehicles used to have "On Government of India Duty" and "Bharat Sarkar" written on it.
The reason why we hear "India" a lot is because Hindi and Sanskrit is not as widely used in official communication. Atleast by the common people. Should we promote Indian languages? Certainly. Should we promote Hindi mainly? I will hesitate at that, I am myself being a Hindi speaker.
Using "Bharat" as the only name for the country will be not the best thing to do. Its best not to shake the boat here. It has been settled matter for a very long time. Let it remain like it. Insist that in ALL official communications, English and an Indian Language name should always be present. Indian Languages should be one of the 14 official languages. Depending upon state in which a document is being made/used, it should bear the language most used in that state. Internationally, English and Hindi may be used.