Well, there is also the distinct possibility that without anything like modern D A profiling no one acknowledges anything. Perhaps Rasputin finds an unfortunate end, but if he does it would be entirely unofficial. I'd guess that there would be just enough evidence of the whole thin to produce endless conspiracy theories.
Right. A good role model was the Czars cousin (second I think),
http://en.wikipedia.org/wiki/Marie_of_Romania
She had a child out of wedlock in Romania and was said to be screwing everything that moved (of high station anyway). Prime ministers, generals, etc. When she blurted it out that the child would be her husband's they clucked but the family covered it up just as you mention. At least it was not the heir apparent first male child. Maybe she sterlized after that, but the frigging in the rigging did not stop and only had begun (circa 1913)!
Queen Mum of UK (George V wife) dispised Marie by 1920, despite the British connection (granddaughter of Victoria), for saying something to the newspaper press like Mae West's statement of needing a man every night. Quite a bitter pill for the old establishment to deal with.
In Russia they had another way to deal with it, also behind the scenes. Rasputin would be dead some how, the child rejoiced as a new source of blood for the Romanovs, and hush is mush. The bigger problem is if the only other male a sickly child (who had a severe hemophilia, which means he bleeds very easily and a lot for vampires or under any other pretext) some how dies, and the newborn is male.
Tongues would wag, putting it mildly. IOTL circa 1916 it was considered an honor that Rasputin would cuckhold a member of the court of somewhat lesser rank, just not ever the head dame, in part because only the royal couple knew why Rasputin was around -- he was thought to have magical powers over the heir's hemophila. Rasputin was not a chaste monk putting it mildly with the Ladies In Waiting, though. Also there were also tons of non court people, often of much less rank like the middle class, who absolutely hated Rasputin. The commoners sort of loved him as one of their own who made it to the top, figuratively speaking only.
But a bit ASB, as Rasputin is said to have more care for self preservation than to get frisky with the Empress -- not not impress the empress with the holy loins so to speak. Word did get around on that score. One of the rumors of how Tchaikovsky, the famed composer, died was that he had a gay relationship with a royal, and was discretely put to death.
Have no idea on that one, and we just do not know the murky details of the composer's demise, but since Rasputin met his end by mere rumors AFAIK is now believed by historians to be rumors completely unfounded, the chance is extremely great that he would be killed quickly and quietly as possible with substance in the flesh. He need not be caught flagrant dilicti.
Also of note,
http://en.wikipedia.org/wiki/Catherine_the_Great was believed to have concieved out of wedlock, under threats of death to concieve by the mother of the Czar, who was "anatomically deficient or uninterested husband". The mother took care of the details, probably introducing
http://en.wikipedia.org/wiki/Sergei_Saltykov. She ended up having three children, all illegitimate, one from the future king of Poland. Sort of a behind the scenes 1700's welfare mother. Times were more cloistered, and revolutionary inflamation would be sure to spread rumors. In war, these would be hard to suppress.