However, if you have a Real Parliament, then they, and whomever was Prime Minister, would be dealing with Hitler. If Alexei becomes a figurehead, like the British Monarch, they he would have no say in Foreign Policy. He could advise, but not command.
The thing is, it would be highly unlikely for Russia to make the jump from autocracy to parliamentary monarchy with just a stroke of the pen. Remember that it took hundreds of years, incuding several civil wars and revolutions, for parliamentary monarchy to develop in Britain. During a what-if reign of an Emperor Alexey II, Russia would probably have experienced some of the same conflicts that central European countries such as Finland, Poland, Hungary, Germany, Italy, Romania etc. experienced in the unstable interwar years. I.e. conflicts between parliament and a (strong) executive and often conflict between the official head of state (president or monarch) and authoritarian, populist leaders.
Poland under Marshall Piłsudski is an interesting example of how a democracy born out of Wilson's 14 points of national self-determination struggled with authoritarian tendencies, not at least because Piłsudski himself is interesting, as a conservative statesman once persecuted by the Tsarist authorities (just like Finland's first President Svinhufvud) because he had been a member of Narodnaya Volya and been plotting to blow up Alexander III.
With the parliamentarian systems and leaders often quite weak, it was often up to the heads of state / monarchs (Vittorio Emanuele III in Italy, Carol II of Romania, Boris III of Bulgaria, George II of Greece, Haakon VII of Norway) to deal with foreign threats such as Hitler personally. If they did, they ran the task of embroiling the crown in dangerous party politics, if they didn't, some local Mussolini was more than willing to do it on behalf of them.
That would have been the dillemma Alexey II would have been faced with. I don't think British constutional monarchy above party lines would have been a viable option in Russia. (Proof: Parliamentary government is not functioning in Russia as we speak, despite 25 years of development since the Fall of the Iron Curtain.) If Alexey didn't want to face the hard choices and grimy realities, I think he would have had to return to Byzantine theocracy, i.e. be some sort of powerless monastic puppet in a golden cage, like the last Chinese emperor.
