1. You can do whatever you like. / You can do whatever takes your fancy.
2. You can take whichever one you like. / You can take whichever one takes your fancy.
Hi! I presume you know there is a difference between 1 and 2, in that doing and taking are not the same thing. You can use them interchangeably but they have a different 'feel'. 'takes your fancy' is informal and sweet, 'whatever you like' is more natural and normal. So I imagine an Aunt using 'takes your fancy'. Number 2 I imaginer you offering a cake! Using 'take' twice in a sentence sounds a little repetitive, so you could say 'You can take whichever one you fancy'.
Both are grammatically correct :)