Step 1: What lowers the extinction coefficient.
A transition that is both spin forbidden and Laporte forbidden is very weak.
Step 2: Examine the complexes.
The tetrahedral cobalt and nickel complexes have allowed, intense bands. The octahedral chromium complex is only Laporte forbidden.
Step 3: Pick the weakest.
[Mn(H$_2$O)$_6$]$^{2+}$ is high spin d$^5$, so every d to d band is spin and Laporte forbidden, giving the lowest extinction coefficient.
Step 4: Answer.
\[ \boxed{[\text{Mn(H}_2\text{O)}_6]\text{SO}_4} \]