I think you need a solid diplomatic overture to India in the late 1960s. To a certain extent, what kept India as a Soviet ally in the second half of the Cold War was the fact that Pakistan was a US ally. If the US instead decides to adopt a more evenhanded approach with relation to the South Asia, India might well warm up to the West. You'll probably need to tweak a few things so that some less socialist politicians gain power but it's doable. Socialism and capitalism can coexist- even now, for example, some Indian state governments are formed by freely elected Communist parties.
Perhaps LBJ goes to India just as Nixon went to China IOTL?
"The Union of India is the world's largest democracy and one which must not be excluded from the Free World..." or something of the sort.
You could do this after India gets a bloody nose in the war with China.