Pakistan would say no due to strained relations, and state level Indian governments would pressure the federal government to say no due to anti-Bangladeshi sentiment in India (sort of similar to anti-Central American sentiment in the US with an added Islamophobic element or Ethnic element depending on the region) - appearing to vacillate to Bangladesh over what is basically a dumb symbolic issue would piss off voters in very narrow Indian elections.
I see that the Iranian government has also demanded the diamond in the past. Forestalling the diplomatic difficulty of giving it to Tehran, give it to the current Pahlavi claimant in exile, satisfying no one in power and thus the truly neutral solution.
No Indian politician would want to appear to be appeasing Bangladesh would be slammed as anti-Hindu, anti-Assamese, anti-Tripuri, etc depending on the election, especially over an unneccesary symbolic issue like the Kohinoor diamond. And fundamentally, no one actually cares about the Kohinoor diamond conundrum other than a couple chattering class types like Tharoor. The Kohinoor diamond question is basically a microcosm of the larger issues in North India, Pakistan, and Afghanistan that all those central goverment politicians exacerbated or directly caused during Partition, the 1980s Punjab Militancy, the Kashmir exodus, the KPK insurgency, the Afghan Civil War, the ongoing issues in Kashmir, the various Indo-Pak wars, etc.
Neither will do it as the other nation would try to make it a client country, and it would anyhow fall back into an ethnoreligious civil war like back in the 1990s. Also, it's not just Kashmir - it's Kashmir AND Jammu AND Ladakh AND Gilgit AND Baltistan. There are dividing lines across religion (Sunni vs Shia vs Hindu/Sikh), ethnicity (Pahari vs Koshur vs Migrant vs Balti vs Ladakhi), and caste (Dogra vs Gaddi vs Bakkarwal). If I was negotiating, the only solution would be a full normalization of India-Pakistan relations with cross-border travel restarted like back before the 1971 war, and potentially a SAARC based free trade corridor (free transfer of goods and people from Mazar-i-Sharif to Moreh, from Gilgit to Galle, from Chittagong to Colombo, from Kathmandu to Karachi), but that ain't happening in my lifetime.
There are a lot of complexities involved (you didn't mention the Buddhists, and China has border claims too), but it seems like the locals are pretty fed up with the status quo and have a real "a plague on both your houses" sentiment now. Unfortunately their opinion doesn't seem to matter to the actual powers involved.
Yep! Everyone hates the status quo all over JK/HP/PN/HR, but any other option will inveitabely bring a return of the bloody days in the 1990s and 2000s, when people were being shot on the streets daily, bombs were going off, and every family have one member who died in the larger bloodshed.
My family and the clan is from that region as well, and just about every family friend has a relative that has died. Jammu, Himachal, and Northern Punjab also have some of the highest per capita rates of per capita Armed Service volunteers in India, so every village or town or family will have a Shaheed/Martyr, which only exacerbates the bad blood. For example, when I go to my dad's village, immediately after crossing the Punjab border into our state you start seeing statues and banners in memory of Shaheeds who died in 1999, 1971, 1965, Uri, Pathankhot, etc. Add to that illegal rifles are very common in the region so there is always the potential for bloodshed to occur. There are generations of bad blood in the area and no one will budge an inch (and who can blame anyone on either side to be honest - when family members die it becomes personal).
Buddhists tend to Ladakhi (a Western Tibetan group). They are closely related to the Baltis (a Shia Western Tibetan group), but there is some bad blood between the two groups, but Ladakh has tended to remain relatively conflict free religion wise (Indian Armed Services and the PLA fight there instead, just like the Sikh Empire did against the Qing Dynasty there in the 1840s (https://en.wikipedia.org/wiki/Dogra%E2%80%93Tibetan_War)).
There is generations of bad blood, and a lot of people with small arms and light arms training. Shit gets real very fast.
PS - I am talking about the Indian side as I have first hand experience there. The Pakistani side is actually Jammu, so a different ethnic group from Koshur. Also a lot of army recruiting from that side into the Pakistani Army. The entire Punjab, Pakhtunkwa, and Jammu Kashmir region has always had a very active martial tradition - think of a Cotton Hill type mentality from King of the Hill.
Give it to Bangladesh then.