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}$. 1 If you interpret locally trivial as follows, then the answer is always, though it will actually be a$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}\$.