I'm going to tell a story that I've related to FT in the past because it is spot on topic.

In 1998, I was commissioned by my university to do research on India's pharmaceutical industry as it prepared to enter the (current) era in which IP is treated semi-seriously.

My university was charged with the task of handling all of the mundane details (tickets, hotels, visas, etc), which I have since learned it is best to take care of on my own.

Anyway, my routing was SFO-HKG-SIN-BOM (with a brief side trip to my PEK ex-HKG; I can't remember how I convinced them to allow that).

The night before my flight out of HKG, I met a middle aged lady in a bar in TST (nothing remotely romantic about this, I promise) who asked me about my India visa.

The next morning, I ran off to the India consulate and learned about the 4-5 day requirement about which you speak. There were no expedite options of any sort; I remember this much.

While my visa didn't happen overnight, by calling the San Francisco consulate (to which my application was referred), my loving mother was able to speed up the process significantly. Even though the period in question was plagued by an atypical convergence of Indian, US, and Chinese holidays, iirc, I was still able to leave in relatively short order. I don't recall the exact specifics, but it couldn't have been that long because the hotel I chose in Wanchai -- recommended by the US consulate; think Suzie Wong -- was so terrible that I would have clearer memories if I'd really spent 5 days there.

Upon reading what I've just penned, I suspect you won't find this info very helpful. On the other hand, I'm living proof that the Indian visa system isn't completely heartless. Furthermore, since you're much more likely to find yourself holed up at the Island Shangri-La instead of those wretchid accomodations I spoke of, I have a hunch you'll make it through the ordeal.

Good luck!