We construct an $S_2\times S_n$ invariant full exceptional collection on
Hassett spaces of weighted stable rational curves with $n+2$ markings and
weights $(\frac{1}{2}+\eta, \frac{1}{2}+\eta,\epsilon,\ldots,\epsilon)$, for
$0<\epsilon, \eta\ll1$ and can be identified with symmetric GIT quotients of
$(\mathbb{P}^1)^n$ by the diagonal action of $\mathbb{G}_m$ when $n$ is odd,
and their Kirwan desingularization when $n$ is even. The existence of such an
exceptional collection is one of the needed ingredients in order to prove the
existence of a full $S_n$-invariant exceptional collection on
$\overline{\mathcal{M}}_{0,n}$. To prove exceptionality we use the method of
windows in derived categories. To prove fullness we use previous work on the
existence of invariant full exceptional collections on Losev-Manin spaces.
