 Opposite point charges have no linear restoring force at their midpoint

Place $+q$ at $-L$ and $-q$ at $+L$. For a nonzero point charge $q_3$ between them, the signed force at $x=fL$ is $2k_{e}q{q}_3(1+f^2)/[L^2(1-f^2{)}^2]$. It is nonzero at $f=0$ and even in $f$, so its first nonconstant term is quadratic. The midpoint is not an equilibrium and cannot be treated as a harmonic restoring centre.