In school, IIRC, they said Rasputin would take the child alone into closed rooms for hours and not disclose what was done. I've always wondered if he was transfusing (which is actually not too difficult to learn--even children infuse themselves). He had the wealth of the family at his disposal--he could have paid for the blood and snuck it in by pretending it was some mystical item.
I am not discounting your theory that blood transfusions may have been involved in Alexei's care. However, the likelihood of Rasputin administering transfusions to the prince in the limited time that he was left alone with him, is unlikely. First off, he would have had to take equipment in with him - and i suppose he could have hid that under his robes. But, that would result in contamination and risk of infection. Along with the inherent risks involved with a procedure of this type that would have been used in an 'arm to arm' procedure of that era:
Not only would that kind of procedure be difficult to do by yourself, it would also be time consuming. Blood transfusion technology was rudimentary in the early days of the 20th century - blood typing was first discovered in 1901.
And then, complicate the procedure by the resulting needle marks which would have to be explained, an overly attentive mother, and a child who revealed everything in his world happenings to his tightly knit family...I find the blood transfusion theory to be somewhat weak, even though not discountable.
The most telling incident that did not involve blood transfusions, was the one of the most serious, that occurred in 1912. It is this incident that was supposedly 'cured' by Rasputin's telegram. I find the notion that Alexei's blood disorder to be one that is characterized by spontaneous remission to be a compelling one, especially as there are few accounts of bleeding episodes after that time.
The mystery of the Romanov family and the last prince of Russia, is receiving some closure due to advances in forensics, yet there will always remain the myths and legends associated with that period in history. Interestingly enough, the entire family has now be raised to sainthood in the Russian Orthodox Church. That is quite the distinction for a family who was murdered in cold blood while all their relatives looked on from a safe distance.
What I find interesting about the Watchtower's slant/bias on their interpretation of history surrounding Rasputin, is that they fail to see the irony in their own position towards blood. Just like Rasputin, the WTS has turned their back on the medical community, and instead, has asked their members to put their faith in some old outdated religious belief. If anything, the Watchtower Society is today's Rasputin.