That is an interesting hypothesis, Rebel, and one that had not occurred to me. I will keep that in mind when I am researching. However, I have not run across anything in the numerous accounts that exist of the Royal family and Rasputin's involvement that would suggest that blood transfusions were involved. If they were, I would assume that a blood transfusion would have involved the medical doctors, not Rasputin, and a medical event like that would not have gone unnoticed or unrecorded.
The prince kept a diary and had a servant with him at all times. I don't think Rasputin would have had the opportunity, even if he had the knowledge (which I doubt - Rasputin wasn't very educated) to administer the procedure. And not have the prince reveal it to someone - the family was very close and intimate with their private affairs.
Many of the 'cures' pursued and even the lengths that the Royal couple went to to try to get that male heir was extensive. The Empress even went so far as to raise Seraphim to sainthood in 1903 in her desire for a son. She also, in her intense desire to produce an heir, suffered a false pregnancy right before conceiving Alexei.
Alexandra was committed to producing a son as heir to the throne of Russia, and after giving birth to 4 daughters, the son was seen as a gift from God a 'Jesus' for Russia. Alexei was born in 1904. And, ironically, the prince shed his blood too, just like the mythical Jesus of their Orthodox faith. He bled throughout his entire short life, and at the age of 14, in 1918, he shed his blood along with his entire family, on the ground of Russia. The history of the last of the Romanov dynasty is one of bloodshed, from its beginnings - right down to the blood of the last prince and his entire family - that era is soaked in blood.
The hemophilia gene was passed onto the Russian male heir by his mother, Alexandra, who was the granddaughter of Queen Victoria. The Empress carried horrible personal guilt for passing this trait down to her son and, by all accounts, was a mother extraordinaire to all of her children. In spite of Alexander's shortcomings politically, nobody has ever been able to fault her as a mother - her devotion was boundless to her family.