Maybe there's another reason. What if Obito/Tobi still has a connection to Kakashi's eye. So if Obito's sharingan evolves and he gains the MS then Kakashi's eye would also gain it as well.
If that were the case then I wonder if Obito can see through Kakashi's eye without him knowing it and gained all the jutsu that Kakashi has copied over the years.
well what could explain this is Kakashi blames himself for Obito's death and the only reason he hasn't obtained it until now is because he didn't know about MS.
its a uchiha secret so the first he ever heard of it was when Itachi used it against him and now hes like :
"oh shit i need that power to compete who would of guessed theres another stage"
and it uses up so much more chakra then it would for a Uchiha that he's left hospitalized and almost dead. as stated earlier in the manga if you convert all ur stamina to chakra and use it up well your dead.
I am McLOVIN
Here's another thought. What if when Itachi took Kakashi into that weird alternate world in his mind he was able to peek into Itachi's memories and found out about the Uchiha hidout. Then after Kakashi recovered he went there and found the same things that Sasuke did. Maybe he even found something that not even Sasuke found, which might have been how to activate the MS without killing your best friend.
Maybe if its sharingan vs. sharingan you can do more with it than if its just sharingan vs. a regular ninja.
what chapter is this that itachi says something after
I am McLOVIN
as you can see kakashi had nothing up his sleeves. he could have died but itachi decided to spare him for an unknown reason.