We consider the quantised free Dirac field on oriented and globally hyperbolic ultrastatic slab spacetimes with compact spatial section and demonstrate how a gauge invariant, pure and quasifree state on the C*-completion of the self-dual CAR-algebra can be extracted from the fermionic projector construction of Finster and Reintjes. This state is analogous to the "SJ-state" of the free scalar field recently discussed in the literature. We prove that this state generically fails to be Hadamard. However, we also show how a modified version of the construction, inspired by work of Brum and Fredenhagen, yields states which are Hadamard. We also relate the Hadamard condition to the finiteness of fluctuations of Wick polynomials.