It's Not Reusable Because It's Broken

In a $Li-FeS_2$ battery at room operating temperature,  at the cathode upon first discharge

$FeS_2 {_{(lattice)}}+2Li^{+}+2e^{-}\rightarrow Li_2FeS_2 {_{(amporhous)}}$ ---(1)

and then a second discharge equation,

$Li_2FeS_2+2Li^{+}+2e^{-}\leftrightarrow Fe+2Li_2S$ --- (2)

On recharge, equation (2) reverses,

$Fe+2Li_2S-2e^{-}\leftrightarrow Li_2FeS_2+2Li^{+}$ --- (3)

then,

$Li_2FeS_2-xe^{-}\leftrightarrow Li_{2-x}FeS_2+xLi^{+}$ --- (4)

but on further recharge,

$Li_{2-x}FeS_2-(2-x)e^{-}\rightarrow (2-x)Li^{+}+FeS_n+(2-n)S$ ---(5)

where $x\ge0.8$

With these background materials...

This is $Li_2$ when two unpaired orbits, each from one $Li$, pairs up,

This is the $FeS_2$ lattice, where each member has six neighbors and each lattice unit is attached to another at two points,

At the cathode, the $Li$ with an unpaired orbit pairs up another $Li$ to form $Li_2$ and attaches itself to the lattice of $FeS_2$.  This is an ionic bond.

$Li_2$ replaces one $Fe$ atom around $S$ and the replaced $Fe$ is reduced.

Under lower temperature $21^{o}C$ to $30^{o}C$, the lattice starts to crack. This is the first discharge when a new battery come into use. Upon a second discharge via equation (2), hole forms in the lattice with the formation of $Fe$ and $Li_2S$.  At this point if a recharge is attempted, the amporhous $Li_2FeS_2$ ⁽¹⁾ reforms within the hole in the lattice.  Upon further recharge, we obtain $Li$, but we do not return to the ionic lattice of $FeS_2$, instead we obtain, $FeS_n$ and free elemental $S$. The presence of elemental $S$ causes havoc at the anode.

Structurally the cathode has disintegrated.

Recharging is not a problem at $400^oC$ with a salt electrolyte, the $FeS_2$  lattice (melting point $1,177-1,188^oC$ is reformed upon full recharge.  Heat increases the orbital radii of the members of the lattice $Fe$ and $S$, and makes the $FeS_2$ lattice more malleable but not melting it completely.  At this temperature the $FeS_2$ lattice can accommodate $Li_2$.

Increasing the distance between the ionic elements can also be done using a polar solvent infused into the lattice.  This solvent should not dissolve the compound but space out the members of the lattice as when the
temperature is at $400^{o}C$.  $FeS_2$ is not soluble in water, but water introduced into the lattice when the crystal forms might space themselves between the lattice members and make the crystal soft.  Otherwise a more polar solvent has to be found.  Since, Pyrite is soluble in acids, a good candidate is $HCl$, other acids are physically too big.  $FeS_2$ doped with $HCl$.

$CuS_2-FeS_2$ has wider lattice spacing $\sqrt{3}.(3.966\times10^{-10})$ m, compared to $FeS_2$ of  $\sqrt{3}.(3.38\times10^{-10})$ m.  And with Arsenic $As$ in the mix,  $\sqrt{3}.(3.8488\times10^{-10})$ m.⁽²⁾

If the lattice is intact after the first discharge via equation (1).  The $Li_2$ member are simply hanging in the lattice of $FeS_2$ slightly expanded to accommodate the new member.  And subsequent discharge,

$Li_2FeS_2-2e^{-}\leftrightarrow FeS_2+2Li^{+}$

pucks the $Li_2$ as $2Li^{+}$ after removing the two electrons, from the lattice.

The crux of the problem is to keep the lattice structure of $FeS_2$ intact during the charge and discharge cycle. A backing lattice that holds onto the $FeS_2$ lattice and stretches it to accommodate $Li_2$ but
itself does not part-take in the redox reactions can be the solution.  Stacked layers of the two lattices will then build the bulk of the cathode.  The backing lattice need not stretch the $FeS_2$ lattice if a single layer of the bounded lattice ($FeS_2$+backing) are use.  In this case, the expose surface of the $FeS_2$ lattice allows $Li_2$ to bond freely.  A transport layer that allows $Li^{+}$ to move freely can be sandwiched between two bounded lattices to build bulk.  Many binary metal chalcogenides (compounds with $S$, $Se$ or $Te$) have the $FeS_2$ structure, as do oxides like $CdO_2$, $\alpha-K_2O$, $\beta-Na_2O$⁽³⁾. And "Late" transition metal disulfides (Mn, Fe, Co, Ni) almost always adopt the pyrite or the related marcasite motif.   There are many material options for a backing lacttice, insulated from the redox reactions by a few layers of $FeS_2$ to provide structural integrity.

Note: It is likely that instead of $Li^{+}$ we have $LiH^{+}$.  Since, $H$ after loosing its electron sit inside $Li$, $Li^{+}$ and $LiH^{+}$ is likely to be of comparable in size.

There is no reversing entropy but entropy can be preserved.

References:

(1) In situ Fe K-EDGE X-ray absorption fine structure of a pyrite electrode in a Li/Polyethylene oxide (LiClO₄)/FeS₂ battery environment.  Dana Totir, In Tae Bae, Yining Hu, Mark R. Antonio and Daniel A. Scherson.  Proceeding Symposium Lithium Battery CONF-961040--23

(2)Schmid-Beurmann P, Lottermoser W Physics and Chemistry of Minerals 19 (1993) 571-577
57Fe-Moessbauer spectra, electronic and crystal structure of members of the CuS2-FeS2
solid solution series.

One set of data from reference (2), (http://rruff.geo.arizona.edu/AMS/AMC_text_files/20081_amc.txt) repeated below, suggests, an atomic center to center of $\sqrt{3}.(6.25\times10^{-10})$m.

Pyrite
Oftedal I
Zeitschrift fur Physikalische Chemie 135 (1928) 291-299
Uber die Kristallstrukturen der verbindungen RuS2, OsS2, MnTe2 und AuSb2.
Mit einem Anhang uber die Gitterkonstant von Pyrit
_database_code_amcsd 0017728
5.414 5.414 5.414 90 90 90 Pa3
atom    x    y    z
Fe      0    0    0
S    .625 .625 .625

(3)"Ceramic Materials: Science and Engineering" By C. Barry Carter, M. Grant Norton publisher: Springer Science & Business Media, 4 Jan 2013.