Liking this TL for the most part, though I do have to say, the Russian Revolution here doesn't make much sense -- I mean, the Duma overthrowing the Tsar and demanding his young daughter, not Michael, replace him? The moderate monarchists explicitly looking to drop out of the war, and by way of killing Rasputin? Working with Lenin?
It's confusing, I admit.
The Duma had done what they have done because of the growing fears of Germany's complete domination of the Eastern Front. Russia has managed minor victories thus far, but nothing has broken the 'Grey Tide' from rolling east, flattening Russia's army in its wake. They don't want their entire empire broken up by German captiulation demands, and are trying to peace out of the war like Japan and Britain have. Serbia has fallen, there's no more point to fighting the war for Russia.
Also, they were afraid of Rasputin's influence over the Tsar and his wife, that he was a dark and evil man of some sort, and they were afraid of the complete breakdown of Russia's ability to maintain peace and order because of Germany's hemorrhaging of Russian troops.
It was decided that while Michael or Alexei would have an immediate claim on the throne, they installed Tatiana because of her pre-existing leadership skills, stable health, and youth; plus their desire to create a
reformed Russia, something that won't collapse every few years to revolution.
The Provisional Government sees appointing a new monarch and separating them from the negative influences of the Romanov's past as important--hence why Tatiana was anointed, and the reason why the rest of the Romanovs are running around trying to evade Trotsky's revolutionaries.
Lenin's decision to work with the Provisional Government (and not join Trotsky) stems from some events during the POD. When Lenin attempted to reassert his power in the Bolsheviks in 1912, he got burned harder than he did OTL; and Lenin reconsidered the viability of the factionalist revolution working from one perspective, and acknowledged another method of doing so. Lenin's revolutionary zeal has softened some, as he has revised some of his revolutionary statements to be more oriented towards
taking over from the inside. Kind of like that one Khrushchev quote--"one day they will wake up under communism", etc.
When the Provisional Government asked for his aid, Lenin resolved that the revolution could be completed without the need to upend
everything all at once-- if the ends justify the means, do the means justify the ends, I suppose. Trotsky doesn't see it that way, and believes that only by total revolution, can the people be free; Lenin sees that if the system can be bent to the will of the People, then the revolution has succeeded anyway. If that fails, then violent revolution is the only way to go.