Yeah im not keen on the Mumbai site either. In my mind it should be one of the most populous cities but anyway...
For China your best bet is normally to take out their poorly defended fringe cities. Lhasa, Xi'an and Ghanzghou. I used to find (in the past, bear in mind I have only palyed it once to the end with the recent patch) razing 2 of their cities was enough to cause a collapse. Sometimes you can encourage the Japanese to go to war with them too, which helps.
I suppose now that the Khmer are close by you could ask them too but I found last game they were too friendly with China (which didnt help them when I took and razed Hanoi).
The Khmer have given me another idea too. If around 1000AD (20 turns should be enough if you have 6 - 10 Elephants stationed in Patna) it looks like you will be struggling to hit the pop target you could always send Elephants into Khmer territory to capture their cities for the 1200AD goal. They grow fast due to the amount of food so should (im going by memory of their capabilities at the moment) help a lot in hitting the goal. Of course that means war over akward terrain in which you can only use Elephants and they can use everything against you. You would also have to take between 3 - 6 cities instead of just razing a couple of Chinas.
By the way sometimes after collapsing China you still find Arabia outpacing you.