No, I don't think Bambi is fine with that.
Also I don't think Ako is draining Bambi, the author probably didn't think too much about the various implications of comparing a soul to a pack of ice.
@zokoi19, thank you for the explanation, but your speculation is kind of weak I guess, I thought about that too and it's not click for me.
I know the snow because Bambi is sleep with Ako, but because she is so cold that's why it's affecting his dream. Also if the snow is the implications of Ako draining Bambi life force, why the snow isn't there before cos Ako is already kind of living together with Bambi since ch 1