  In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant
for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of
so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant,
$\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of
$Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots
which is typical of Floer-theoretic invariants of $3$-manifolds.
  In this article, we demonstrate further structural properties of this
symplectic instanton homology. For example, Floer theories are expected to
roughly satisfy the axioms of a topological quantum field theory (TQFT), so
that in particular they should be functorial with respect to cobordisms.
Following a strategy used by Ozsv\'ath and Szab\'o in the context of Heegaard
Floer homology, we prove that our theory is functorial with respect to
connected $4$-dimensional cobordisms, so that cobordisms induce homomorphisms
between symplectic instanton homologies. We also generalize the surgery exact
triangle by proving that Dehn surgeries on a link $L$ in a $3$-manifold $Y$
induce a spectral sequence of symplectic instanton homologies -- the $E^2$-page
is isomorphic to a direct sum of symplectic instanton homologies of all
possible combinations of $0$- and $1$-surgeries on the components of $L$, and
the spectral sequence converges to $\mathrm{SI}(Y)$. For the branched double
cover $\Sigma(L)$ of a link $L \subset S^3$, we show there is a link surgery
spectral sequence whose $E^2$-page is isomorphic to the reduced Khovanov
homology of $L$ and which converges to the symplectic instanton homology of
$\Sigma(L)$.
