Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$

We compute the Floer homology and Seiberg-Witten Floer homotopy type of Seifert rational homology $3$-spheres which fiber over $\mathbb{RP}^2$. We show that they are all $L$-spaces and their Floer homotopy type is a suspension of $S^0$. Additionally, we compute the Ozsváth-Szabó $d$-invariants, or equivalently the Seiberg-Witten $δ$-invariants for such $3$-manifolds. This is done by computing the eta invariant of spin$^c$-Dirac operators associated to spin$^c$-connections covering the adiabatic connection, a certain metric connection distinct from the Levi-Civita connection. It turns out that this eta invariant involves a contribution given by the eta invariant of an orbifold pin$^c$-connection on the orbifold base of the Seifert fibration, which we also compute.