Some issues:
1. "Constitutionality". It's becoming a kind of mantra in this thread. We have to accept that George V could not have behaved otherwise because "he was a constitutional monarch".
Let's check the facts:
There's a government in Russia that has been recognized by Britain and that is an ally in the ongoing war. A representative of that government, the Minister of Foreign Affairs Milyukov, officialy requests Britain to grant asylum to the Imperial Family. The Prime Minister, the head of the Cabinet, which has most information about the situation in Britain, decides to comply with that request of an ally. And then, the Constitutional Monarch George V panicks, and based in some letters that he has received, intervenes to block the whole matter via an informal contact through his secretary with Foreign Secretary Balfour.
Is that what a Constitutional Monarch is supposed to do? Block political decisions taken by a Prime Minister? Bypass the whole political system? Lobby Cabinet ministers in his favour? I am not an expert, but I think that isn't very constitutional.
2. "It would have dented his popularity or the popularity of the institution that is the British Monarchy". If a Monarch is called to do something callous (abandoning a close relative he called his friend fell in extremely dangerous circumstances), there's something very wrong with the system. Maybe he should challenge the system or give up his crown, pack his things and settle in Madeira or other island with a good climate.
3. "The chances of the Imperial family actually getting out of Russia were still pretty remote". That's a bogus argument.
A man is involved in a car crash in a small town and taken to the local hospital with very serious injuries. The only surgeon available refuses to operate him. The wounded man is put again in the ambulance and taken to a further hospital. He dies on the way. The autopsy determines that his chances of survival if he had been operated were small, around 5%. Even so, would the surgeon who refused to operate him get off scot-free? Wouldn't he incur at least moral censure?
And now, let's return to the Russian Imperial Family and George V. Maybe all of them could have been saved. Maybe only the children could have been saved. Maybe the Grand Duchesses (but not Alexey) could have been saved. I am not going to assign probabilities to each of those scenarios. You can make the chances as low as you want. The fact is that George V DID think that there was a real chance of taking them out of Russia to safety in Britain. That is why he acted to block it. If he had thought that was something purely hipotetical that would never materialized he would not have panicked.
"What does remain certain is that the King, by persuading his Government to withdraw their original offer of asylum, deprived the Imperial family of their best, perhaps their only, means of escape." Kenneth Rose, King George V