It seems we cannot choose such a center if we want it to be preserved by isometries, that is, if for any isometry $i:\mathbb{R}^2\to\mathbb{R}^2$ we want the center $p_F$ of any compact convex $F$ to satisfy $p_{i(F)}=i(p_F)$.


This is because if the center satisfied that, as the triangle $T$ with vertices $P_1=(-1,0)$, $P_2=(1,0)$ and $P_3=(0,1)$ is invariant respect to the reflection respect to the $y$-axis, we have $p_T=(0,y)$ for some $y$. If $y=0$ then we run into the counterexample mentioned in the question. If $y>0$ then consider the triangle $T_2=(5,0)-T:=\{(5,0)-q;q\in T\}$. Then $d_H(T,T_2)=10$, but $d(p_{T},p_{T_2})=d((0,y),(10,-y))>10$.