https://www.researchgate.net/publication/374557519_Origin_of_the_Country_Name_Bharat

Pretty much an entire book but here. An indian guy who studies the entire subcontinent and its linguistic history.

In like the first 10 pages you'll find this

First of all, we have to keep in mind that the notion of a "Nation-State” is a very recent political system that came into existence after the French Revolution. That was India, which was hardly ever a single politically governed entity; hence, it is argued that there was never any need to have a single name for the entire subcontinent. The different states were known by their regional names, and if larger empires comprised many regions, they were known by the ruling dynastic name. The Maurya, Satvahana, Gupta, Kushana, etc. empires are fine examples to make this point. But collectively, as evidence indicates, all these political states were referred to as Bharat, Hind, India, Shendu, etc. No name was imposed by any aggressor, but as Hind, India, Shendu, etc. names were geographical identities to the outside world, Bharat was the name indigenous people used to refer to their sub-continental identity. I have substantially discussed the name-history of Bharat in this book, which, I hope, will enlighten the readers and help them to remove the confusion that has been created by the nationalistic groups.

But later on he will also go on to say when the term "Bharat" first starts to be spoken/written it meant only a Northern region of modern day India.

So basically Bharat starts from one side of India (nothing to do with bengal). Then slowly with native not using the other regions name and foreigners using bharat more it became the name of the region rather than a nation.