Why we can't use non-deterministic turing machines in this case?
Your reasoning is not wrong. But recall that decidability requires TM machine halt with YES if $\exists w \text{ such that } w^4 \in L(G)$ or NO if $\nexists w \text{ such that }w^4 \in L(G)$. In other words, the TM always halts.

In your approach the TM machine has to guess infinitely many strings and check if their forth power belongs to $L(G)$. A nondeterministic Turing machine cannot check infinitely many strings in finite amount of time. 

Decidability problems efficiently solved by non-deterministic Turing machines are also solved by deterministic Turing machines. In other words, their computational powers are equivalent.