Tamil Hindu here, Lord Ram isn't popular in my state. In fact many Dravidians consider Ramayan as an act of aggression by outsiders against their lands/people. What I mean to say is that Hinduism can't be defined on the basis of one avatar or one epic, each place has its own version, interpretation and inspiration, even deities. Hinduism is an umbrella term that creates a broad framework for a multitude of faiths and beliefs to co-exist. A forest dweller or some isolated tribesman may have his/her own distinct animistic faith and that too is brought under the fold of Hinduism because of the flexibility Hinduism offers. There is no central book or holy place, or defined set of rules like in Islam or Judaism.
Now what BJP/RSS/VHP/BD practice is entirely different. They are followers of Hindutva (founded in early 20th century) which is a toxic mix of politics and religion with fascism as an underlying feature. The ideologues of Hindutva were inspired by the fascist movements of Europe. Example Golwalkar, one of the leading figures of RSS and a hero for Hindutva forces cited Mussolini as his inspiration. RSS is the brain and BJP is the political front. They want cultural homogenization be it with respect to language or religion (within Hinduism as well), with hegemony of upper castes. PPers mostly see only the religious angle but the Sangh also has waged a soft war against non-Hindi language speakers and other interpretations of Hinduism. For instance the cultural aggression in Bengal where they want to supplant the pre-eminent Shakti (Durga, Kali) and introduce Ram, it reached a tipping point during the election campaign and is still seen as cultural aggression by many there. Hindutva is not an extreme version of Hinduism, IMO it is an enemy of Hinduism and a direct threat to the religion which has always been about accommodating various strands of the umbrella in a non-confrontational way. Religio-cultural homogeneity isn't what Hinduism is about, Hinduism is open to considering Buddha as an avatar and co-opting Sufi saints, Hinduism places no bar on praying to Jesus/Mary for it sees divine in holy figures everywhere.
How BJP has captured political power can be left for another thread but it is a confluence of many circumstances each of which will have to be analyzed in detail. BJP's Hindutva hasn't touched Kerala or Tamil Nadu, they have never won a seat here. It won't be right to generalize Hindus as some PPers mistakenly do, Tamil Nadu has 88% Hindu majority, 75 million population in the state. Andhra is 90% Hindu and again BJP has made no inroads, the present CM is a Christian who swept the polls. There are historical reasons and regional politics which have stopped Hindutva in these states, so one can't take BJP as an example and get insights about the Hindu mind. Many North/West/Central Indian Hindus may be enamored by BJP but this may be a temporary phase, we don't know the future. I guess almost all countries have gone through such phases in their history only to recover, pendulum swings one way or the other and of late not just in India but RW fascist forces have gained significant ground even in Europe and USA.
Coming to your point about Kashmir, the Tamil and Malayali Hindus will have a different take than someone in Haryana or Uttar Pradesh, varies from state to state. What I have observed here, most aren't happy with the development, there have been protests and also a sense of fear about what is next to come (along with Kashmir, Tamil Nadu is seen as a final frontier by the Sangh). So while Tamil Hindus aren't ardent followers of Ram like in some other states (your OP) most do not subscribe to this misadventure by the national government. Our popular leaders have made their points over this and most of us agree with them.
<blockquote class="twitter-tweet" data-lang="en"><p lang="en" dir="ltr">I urge the President of India to not precipitate the situation and not take any further steps in this regard until a democratically elected Government is formed there. <br><br>The DMK stands with its Kashmiri brothers and sisters and will oppose any assault on federal structure.</p>— M.K.Stalin (@mkstalin) <a href="https://twitter.com/mkstalin/status/1158352492855353344?ref_src=twsrc%5Etfw">August 5, 2019</a></blockquote>
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