The Void Let Them In

The octahedral void exposes $S$ in $FeS_2$ on the surface and made it vulnerable to a free $Li_2$.  One $Fe-S$ ionic bond is broken with the formation of one $Li_2-S$ bond.  This one broken bond out of six with a big $Li$ jammed in the lattice causes the lattice structure to crack, and data from X-ray absorption suggests that $Li_2FeS_2$ is amorphous.  At room temperature, on recharge the lattice does not reform and a different sets of redox reactions applies which results in the precipitation of $Fe$ and $S$.  Both are undesirable in the operation of the battery.

It's the hole's fault, we need a bigger hole.

A mix of smaller "late" transitional metal might allow $Fe$ to open up.  Cobalt $Co$ is a suitable candidate.
And $CoS_2$ (beta form) can serve as a backing lattice to keep $FeS_2$ structural integrity.  But $Co$ is further down the reactivity series, it is likely that $Co$ is reduced by $Li$ before $Fe$ in which case we would use $FeS_2$ as the backing lattice and use $CoS_2$ to receive $Li_2$.  $CoS_2$ will insulate $FeS_2$ from $Li_2$ while $FeS_2$ provides structural integrity.

Another big "late" transitional metal is $Mn$ with crystal radius of $0.81\times10^{-10}$m.  $Mn$ is more reactive than $Fe$.  In place of $Fe$ as the backing lattice and dopant for a reactive $CoS_2$ layer might open the lattice ($CoS_2$ doped with $Mn$ lattice) further and make $S$ more accessible to $Li_2$.