We have a similar problem, but in reverse. It's a UK website that needs to rank in Asia but not UK.
We've tried a few things - shifting the server made no difference (nor the speed benefits) and geo locating on webmaster tools was pointless as we have a 5 country target market. Targeting cities again didn't help because the number of location targeted searches was sub 1% for the relevant keywords.
So there were two things that we did that seemed to have some benefit. First we targeted some non valuable but relevant keywords with high search volumes (2 orders of magnitude more than our real targets) and got a high ranking for that. Then put our effort into getting this traffic to convert to the non-ranking pages.
Secondly because there was non substantial difference in EU and Asian ranking for our keywords we focused on ranking well in Europe and then just filtered out the EU and US traffic and we saw benefits.
What I'd do here in your case is focus on the Indian market to get lots of relevant back links. It should be easier than the US in someways and take advantage of the size of the market and it's lack of sophistication to increase the volume of keywords. If websites are anything like they are in Asia there is a lot of crap design and zero optimisation. So it should be possible to put together an outreach package that generates lots of low DA links that are very relevant for lower costs than in the US.
So for example if you were looking at outsourced accounting I'd build link bait that appeals to Indian accounting firms to get the links to your key pages - but optimise them for US consumption.
Of course all this depends on your market and keywords but I hope it's given some ideas.