  Let $A$ be an abelian variety over a number field $F$, and suppose that
$\mathbb Z[\zeta_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of
unity $\zeta_n$ of order $n = 3^m$. Assuming that the Galois action on the
finite group $A[1-\zeta_n]$ is sufficiently reducible, we bound the average
rank of the Mordell--Weil groups $A_d(F)$, as $A_d$ varies through the family
of $\mu_{2n}$-twists of $A$. Combining this with the recently proved uniform
Mordell--Lang conjecture, we prove near-uniform bounds for the number of
rational points in twist families of bicyclic trigonal curves $y^3 = f(x^2)$,
as well as in twist families of theta divisors of cyclic trigonal curves $y^3 =
f(x)$. Our main technical result is the determination of the average size of a
$3$-isogeny Selmer group in a family of $\mu_{2n}$-twists.
