If you interpret locally trivial as follows, then the answer is always, though it will actually be a $Cosk^{n-1}G_{\bullet}$ principal bundle.

The interpretation of locally trivial which I want to suggest is that $P_{\bullet}\rightarrow X_{\bullet}$ is locally trivial if there exists a hypercover $c:U_{\bullet}\rightarrow X_{\bullet}$, such that $c^*P_{\bullet}\rightarrow U_{\bullet}$ fits into a pullback diagram (which I'll describe since I can't figure out how to draw it in MathJax).

The diagram expresses $f^*P_{\bullet}$ as the pullback of the universal $G_{\bullet}$ bundle $WG_{\bullet}\rightarrow \bar{W}G_{\bullet}$ along a cocycle $U_{\bullet}\rightarrow\bar{W}G_{\bullet}$.

With this definition of locally trivial, we can see that $Cosk^n$ preserves local triviality as follows: namely, $Cosk^n$ preserves limits (it's a right adjoint), so applying it to the pullback square, we get another pullback square. Now, a calculation in the definitions of $W$, $\bar{W}$ and the combinatorics of simplices shows that $Cosk^n\bar{W}=\bar{W}Cosk^{n-1}$ and likewise for $W$. Cosk^n\bar{W}=\bar{W}Cosk^{n-1}$. Working backward through the definitions, we see $Cosk^nP_{\bullet}$ is a principal $Cosk^{n-1}G_{\bullet}$ bundle over $Cosk^nX_{\bullet}$.

If you interpret locally trivial as follows, then the answer is always, though it will actually be a $Cosk^{n-1}G_{\bullet}$ principal bundle.

The interpretation of locally trivial which I want to suggest is that $P_{\bullet}\rightarrow X_{\bullet}$ is locally trivial if there exists a hypercover $c:U_{\bullet}\rightarrow X_{\bullet}$, such that $c^*P_{\bullet}\rightarrow U_{\bullet}$ fits into a pullback diagram (which I'll describe since I can't figure out how to draw it in MathJax).

The diagram expresses $f^*P_{\bullet}$ as the pullback of the universal $G_{\bullet}$ bundle $WG_{\bullet}\rightarrow \bar{W}G_{\bullet}$ along a cocycle $U_{\bullet}\rightarrow\bar{W}G_{\bullet}$.

With this definition of locally trivial, we can see that $Cosk^n$ preserves local triviality as follows: namely, $Cosk^n$ preserves limits (it's a right adjoint), so applying it to the pullback square, we get another pullback square. Now, a calculation in the definitions of $W$, $\bar{W}$ and the combinatorics of simplices shows that $Cosk^n\bar{W}=\bar{W}Cosk^{n-1}$ and likewise for $W$. Working backward through the definitions, we see $Cosk^nP_{\bullet}$ is a principal $Cosk^{n-1}G_{\bullet}$ bundle over $Cosk^nX_{\bullet}$.