This answer is without specific knowledge of Indian cultures or languages, but I can offer some perspective on rap compared with other poetic/oral poetry traditions that might help you.
In short, I think you can bring the essence of hip-hop/rap to India, but it will take on some characteristics of Indian tastes to become popular and to fulfill your criteria of classical content.
Background on rap vs. other poetic traditions
American rap music is only the latest manifestation of oral/spoken-word poetry: it's not sung, and it is rhythmic and follows metrical rules (though determined more by a beat than an established metrical scheme). Interestingly, what makes rap controversial is actually nothing new in world history-- just in recent English traditions.
Rap differs from recent-historical poetic traditions in English because it does four things:
- It tends to be set to a driving beat pattern (almost always in 4/4) with
simple musical riffs.
- It eschews flowery, "poetic" lyrics in favor of contemporary speech
- It favors themes and content of a personal, immediate nature to the exclusion of
philosophical and impersonal themes and literary references. (Obviously those are extremely broad generalizations with many counterexamples.)
- It often includes explicit language and taboo-breaking themes of
sex, drugs, violence, and confrontation.
(I'll come back to #1 after I address the language and content.)
On the language and themes of rap
Only in the past century has English experienced a sort of revolution against using deliberately beautiful, intricately crafted turns of phrase in favor of using simple, straightforward language. It's a stylistic phenomenon that has made writing more like common speech and also allowed common speech to become even plainer. Literary-- and even biblical-- allusions and symbolism are too florid and therefore have fallen out of favor. (This trend is also antithetical to the writing traditions of many other languages, including, I gather from your question, those spoken in India.) This style is ubiquitous. Even the more lyrical contemporary poets (Bob Dylan, e.g.) use simple structures paired with evocative imagery and irony rather than dense intertextuality and lush language. Pop music lyrics are usually extremely basic. Rap, too, follows the contemporary English plain-speaking aesthetic. So what makes for good* rap lyrics? Usually clever concepts and wordplay, but not oblique phrasing (or literary allusions).
Rap tends to favor directness in content as well, probably because of both the English linguistic culture and the society whence it comes. You don't need an English degree to be a poet, but you might want one to be respected among the literary elite that dominate traditional poetic scholarship and criticism. By contrast rap music is rather egalitarian and deals with the here-and-now, usually with very personal concerns and visceral emotions. It is by and for an American culture (black Americans**) that traditionally has not been given equal access to the academic side of poetry. Of course there are many examples of rappers that do explore other themes and that do tap into literary and poetic traditions.
You describe rap as having as "dark" themes, and many of them are-- because rappers are exploring very real worries and tragedies in their own lives or those of their friends and neighbors. It also gets into frivolous or bon vivant themes-- having lots of money, going to amazing parties, getting with hot people-- that serve as a foil or antidote to the dark times.
Rap has been surrounded by controversy because it doesn't shy away from confrontation or taboo subjects and language. But you can look at the history of "great" (lauded) literature and poetry to find many parallels. Glorification of gory, explicitly described violence? Read the Iliad. Extremely foul insults to the poets you're beefing with? Catullus. Gushing odes to sex? (Also Catullus! Did you know he gets taught in all the fun Latin classes?) Also see the troubadours. Substance abuse? Lewis Carroll, Arthur Conan Doyle, and every musician from 1968-1978. Heck, Shakespeare was downright rude, but it doesn't come across so easily when all the expressions of nastiness have changed. So why is rap such a big deal? Mainly, US society still retains some Puritanical and Victorian hangups (less so in the age of the internet, and after 40-odd years of hip-hop in the mainstream) about what is polite and acceptable in public contexts. Sadly, I'm sure that racism also factors into making it a particularly divisive genre.
So, how to translate the style and content of rap to Indian languages and cultures?
I think there could be a version of rap that makes its way into Indian mainstream, but there will have to be some adjustments. You can't have your lyrical, ancient, safe stories and also your edgy, conversational, existential ones all in one bite. (But you could have a genre that does all of it piecemeal.) I think there's a separate question here of cultural conservatism, addressed in brief at the end.
Either the content will be at least as controversial and taboo-breaking as American rap has been, or its thematic potency will be diverted into more of the same safe content that is already popular in India. If songs about ancient history and religion are hits today, why not rap about them as well? Or if mindless partying is a popular topic, there is no reason it couldn't also be celebrated in rap. But if there are groups that are as marginalized as black people have been in the US (and I know there are), they might want to explore in music the immediate concerns and outrages in their lives rather than tackling sweet love songs or heroic themes. (They could be doing this already with a different style of music, I don't know, or perhaps they could adopt both the style and substance of rap.) What taboos would be broken in Indian rap to have an effect similar to the controversial elements of American rap?
Perhaps you can bring in the wordplay of rap into Indian languages, but it may not fit conventional tastes. We rhyme in English because our words don't already have meaningful suffixes that naturally rhyme; rhymes are clever. In romance languages rhyme schemes are disdained because they are so obvious. English goes through periodic phases of lauding or despising puns. What makes for clever wordplay in your target languages? Go for that rather than copying the method of wordplay from English rap.
Why not have extremely poetic, oblique language delivered in a new aesthetic as rap? If what you want is a conversational tone to the lyrics, you can introduce that in any genre-- but again, if there is no taste for it you won't get the popularity and mainstream appeal that you are hoping for. The important thing is that the new genre resonate with some segment of the population.
Poetry in any language is affected by the patterns of stress or length that predominate in the spoken language. Therefore some metrical schemes are a better natural fit to some languages than others. English poetry is dominated by iambic pentameter because we tend to alternate between stressed and unstressed syllables in our natural speech.
The advantage that lyrics to music have over unaccompanied poetry is that the beats define the structure and the writer is then free to play with the meter. Contemporary English music takes full advantage of this and often ends up with lyrical metrical schemes being defined by some combination of genre conventions, the time signature, the melody, and the intended meaning of the words themselves. You'll still find that the natural stresses in the lyrics must align well with the points of emphasis in the music.
Rap tends to operate strictly in 4/4 time, beats tend to receive heavily stressed syllables, and edgy patterns can be introduced with syncopation, deliberate delays, etc. Different artists tend to favor certain structures. You might try doing some metrical analysis: look at the lyrics of a rap song and figure out where all the beats fall, then think about them statistically. What rhythms predominate? Where do the stresses fall? (Now, look at the top 100 rap songs of 2018 and crunch numbers...)
How well would those patterns align with Indian languages and poetry? I'd guess that it's not a perfect fit. If you want to bring rap metrical structure to a new language you can probably make the percussive 4/4 work, and it would certainly signal the genre. However, you would probably find that different rhythmic patterns are more successful in the new language, the metrical trends would shift, and you would end up with a somewhat different (perhaps very subtly different) aesthetic to the music.
On whether Classical music will evolve
I can't speculate much on Indian musical traditions, but generally if your music is "classical" it is deliberately preserving older traditions and is very conservative. You can take the content or form out of the classical and put it into a new genre, but you probably won't have much success going the other direction. That's ok! We have many different styles of music for a reason. But if the culture as a whole is very conservative you won't get as much buy-in for new musical influences and genre experimentation. American music was influenced by many different cultures and the deliberate exploration of new material and instrumentation, and new styles emerge all the time because there is support in the culture for it (though there are some people who listen exclusively to classical music).
**As @ChrisSunami points out in the comment below, some of the most influential and innovative music of the past century has come out of black communities in the U.S., who are largely the descendants of people from all over West Africa who were enslaved and separated from their communities. I would be interested to have a better understanding of how having access and close ties to one's cultural traditions, or being separated from them, might drive groups of people to be more culturally conservative or to develop innovations. Does musical innovation happen as a product of syncretism (the blending of cultures that occurs when populations mix), or to fill a loss when people leave a place where their culture is concentrated and firmly established, or both? In India, where religious and cultural traditions date back millennia (and where there can be conflict along some religious lines), I am not surprised that there is such support for traditional music.