Let $(H_n)$ be a sequence of different finite dimensional complex Hilbert spaces, $A_n \in B(H_n),tr(A_n) \to 0(n \to \infty)$,but the norm of $A_n$ does not converge to 0,where $tr()$ is the standard tracial state.
Can we construct a sequence of operators $(P_n)$ such that each $P_n \in B(H_n)$ , $\|P_n\| \to 0 $and $tr(P_nA_n)$ does not converge to 0?
Yes.
Take $H_n = \mathbb{R}^2$ for all $n$, and $$ A_n = \begin{bmatrix}a_n& 0 \\ 0 &-a_n\end{bmatrix} \quad P_n = \begin{bmatrix}b_n& 0 \\ 0 &-b_n\end{bmatrix} $$ where $(a_n)_n$ and $(b_n)_n$ are two sequences we will choose in a second. Then clearly $\operatorname{tr}(A_n) = 0$ for each $n$, and $||P_n|| = |b_n|$, so if we choose $(b_n)_n$ such that $\lim_{n} |b_n| = 0$ then we satisfy your assumptions.
Now $\operatorname{tr}(P_nA_n) = 2a_n b_n$, so it is enough to choose $a_n = (b_n)^{-1}$ to have $\operatorname{tr}(P_nA_n) = 2$ which clearly does not converge to zero.