Assuming an Iranian state manages to maintain long term control over part or all of India, having Persian be an upper-class language is possible-- in OTL, Persian still managed to become an upper-class language in Indian Muslim states despite migrants from Iran being a small portion of the upper class (which also included Turks, Afghans, and native Indians) and an even more miniscule portion of the population at large. Likewise, having Iranian administrators would not seem to represent a problem given that they enjoyed such status in OTL's Indian Muslim states for centuries.
However, what you're asking for is rule of India by a state based in Iran, not an Indian-based state influenced by Iran and Iranians-- and this is much more difficult.
View attachment 447733
Let's assume these are the rough borders of our mega-Iran.
There are a number of ways in which India can be cut off from Iran:
- An Afghan rebellion erupts and captures the Khyber Pass (red marker), cutting off a land route to India. This rebellion is likely sparked by excessive Iranian interference in local affairs, which also influenced the OTL rebellion of the Hotakis in Kandahar. Given that the same rebellion actually destroyed the Safavid empire, and that another Afghan rebellion-- that of the Durranis-- ended with the entire Afsharid treasury stolen, I don't think mega-Iran would do much better.
- Disquiet among the Baluchi or Khorasani tribes. This was a major obstacle to the early Qajars' efforts to govern east and southeast Iran, and while that problem was awaiting resolution the overland flow of pilgrims from Indian Shiites to places in Iran nearly stopped completely. With their ability to disrupt transport and governance, angered Baluchis can easily throw a wrench in the works of mega-Iran.
- Weakness in the Iranian navy, whether due to budget shortfalls or defeat in war. This can worsen the effects of #1 and #2 as well.
Even if the Iranian government maintains its road to India, at the end of that road you're facing Sikh resistance in the Punjab, raids from the Marathas or Nepalis, etc. The British East India Company won against these challenges in large part due to its independence from London-- its governors in Calcutta, Bombay, and Madras gathered information quickly and had wide decision-making powers. If Iran wants its Indian administration to have a similar dynamism, it will have to delegate to it the freedom to coordinate the conquest of India based on its own expertise. This amounts to quasi-independence. By this time, the Persian-EIC might even have the support/grudging acceptance of local civilians/locally-recruited troops and seize some pretext to formally split off from Tehran's meddling. If the locals are hostile, though, an alt-Sepoy rebellion may bring the PEIC down before Iran can intervene.
Extremely peripheral bits like Bengal may also be impossible to hold for long-- Bengal-based elites drew on its population and resources to defy Delhi Sultanate and later Mughal rule for centuries, and even when subdued they usually enjoyed a great deal of local autonomy.
Finally, this mega-Iran may not remain Iranian for very long. In OTL, the Ghaznavids were the first Islamic dynasty to enter India's heartland. Meanwhile, their capital gradually shifted from Ghazna to Lahore, at which point they not only got most of their revenue and population from India but were also based in India. 700 years later, the Nizari Shi'i Imamate decided to move from Qom to Kerman to be closer to its Indian followers, the main source of its revenue. The Iranians may soon find that the Persian-EIC is so wealthy and influential that it dominates Iran's strategic and policy planning, like a tail wagging the dog. At that point, Iran is essentially the junior partner in a megastate that sees India as its priority.
***
Of course, this is all based on the premise that the "Persia" which conquers India is Islamic and Early-Modern. I'd be interested in scenarios that look at a medieval Islamic conquest (maybe the Ghaznavids win against the Seljuks and get to keep both their Indian and Iranian holdings?) or a pre-Islamic Achaemenid or Parthian invasion.