In your inequality you need to evaluate the $\pi$'s somewhere! 
Unless I am mistaken, unpacking [Fell (1962), Theorem 2.2 and Remark following](http://www.ams.org/mathscinet-getitem?mr=150241), $[\pi_j]\to[\pi]$ means that for every choice of an $\varepsilon>0$, a compact set $K\subset G$, an integer $n$, and vectors $u_1,\dots,u_n\in\mathscr H_\pi$, there exist a $j$ and $v_{j1},\dots,v_{jn}\in\mathscr H_{\pi_j}$ such that 
$$
\left|\langle u_i,\pi(g)u_i\rangle-\langle v_{ji},\pi_j(g)v_{ji}\rangle\right|<\varepsilon
$$
for all $i\in\{1,\dots,n\}$ and all $g\in K$. In short, positive-definite functions associated with $\pi$ are *uniform-on-compacta* limits of positive-definite functions associated with the $\pi_j$.
(See also [Dixmier (1977)](http://www.ams.org/mathscinet-getitem?mr=458185), 18.1.5 for groups and 3.4.10 for general C$^*$-algebras.)