The POD for this would actually need to be much earlier than Indian Independence in 1947. You'd need to lay the groundwork for it in the 1930s around the time of hte 13th Dalai Lama's death in 1933. HHDL was trying his hardest to modernize Tibet, going so far as importing Japanese military advisors and sending boys off to England to learn modern industrial techniques. Tibet gained semi-recognition because of its relationship with the British, culminating in Charles Bell's appointment as a British ambassador to Lhasa, and resulting in British recognition of Chinese suzerainty over Tibet (as opposed to sovreignty) into the 21st Century.
The 13th Dalai Lama's quest to modernize Tibet was not unchallenged. At one point the monks of Sera monastery stormed the Norbulingka and shouted offenses (and literally shat all over the garden) at the Dalai Lama. Sera was the epicenter of the Tibetan traditionalism and they were opposed to the DL's formation of a standing army, of decreasing monastic taxes, and modernizing Lhasa's infrastructure and education. The new Tibetan army was employed and successfully laid siege to Sera, capturing the monks and the leaders of the revolt. (At least I'm pretty sure it was Sera. If not Sera, than Drepung. But definitely not Ganden since Ganden Monastery was the source of most of the Dalai Lama's official power. The government of Tibet from 1642 - 1959 was actually called "Ganden Phodrang," or "Palace of Ganden.") Once the DL died in 1933, the lynchpin for Tibetan modernization was gone and the traditionalists took over, torturing what modernists were left, closing the western-style schools HHDL had opened, and destroying or neglecting the institutions they built (like the army and the infrastructure).
Strengthen and focus the 13th Dalai Lama's efforts at diplomacy with Britain, strengthen economic connections with India, position the modernists to take power in Tibet, and India could be willing to go up to bat for Tibet. But I think by 1947, it was way too late to change anything.
I've actually been floating around the idea of writing an Independent Tibet TL, hence why I've been thinking about this.