No need Kurama chakra, he was ok the last time we saw him.All he had to do was to rest a bit and think about strategy.
Since he got Kurama's chakra before warping Obito to kamui dimension, he only used 1 kamui, 2 raikiri and few genjutsu so he still have a lot of chakra.
Furthermore it's not possible to receive Kurama's chakra while he's in kamui dimension...Chakra flows moves and need direct contact with chakra as explained in last chapter, in Kakashi's case, Kurama's chakra can't go to kamui dimension...
Apart if Kakashi opened a kamui gate when Naruto transfered the chakra...