  Given a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S$, we
provide criteria to determine when uniqueness of factorization into irreducible
$\mathcal{F}$--invariant representations holds. We use them to prove uniqueness
of factorization when the order of $S$ is at most $p^3$. We also describe an
example where the monoid of fusion-invariant representations is not even
half-factorial. Finally, we find other examples of fusion systems where this
monoid is not factorial using GAP.
