If the government had decided to rescue the Tsar, wouldn’t the King be required as a constitutional monarch to follow the advice of his ministers ?
What I find odd about this and other similar versions of the story is that they seem to imply that George V actually governed the country and had a final say in foreign affairs or military operations, which was definitely no longer the case in the UK in 1917. I don’t dispute that George V may have been personally opposed to granting asylum to the Tsar , but , even if he had expressed his views in private to the prime minister, it would have been ultimately the government’s call to organize a rescue operation or not.
I also tend to think that the assumption that the Tsar’s presence in Britain would be a risk to the British monarchy is overblown. First, it is unclear how long the Tsar would stay in Britain. Most likely in my opinion he would have moved eventually to a neutral third country. Second there was no credible domestic threat to the monarchy in the UK and even less so after victory in World War I and the introduction of universal suffrage following the war. Of all European countries at the time, the UK ( excluding Ireland of course) was one of the least likely to experience revolution or even political upheaval. I don’t think the Tsar’s presence in the country would have changed that in any significant way.
Clearly the presence of the Tsar and his family in the country would put the UK in conflict with the new Soviet regime in Russia , but the Soviets were not a direct threat to the British Empire ( yet) at that time and, in any case, the British government was supporting the counter-revolutionary White Army anyway, so I don’t think there was too much concern in London about antagonizing the Soviets.