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.