  In ``Knots in lattice homology", Ozsv\'ath, Stipsicz, and Szab\'o showed that
knot lattice homology satisfies a surgery formula similar to the one relating
knot Floer homology and Heegaard Floer homology, and in previous work, I showed
that knot lattice homology is the persistent homology of a doubly filtered
space. Here I provide an iterable version of the surgery formula that, provided
the initial knot lattice space with flip map, produces a space isomorphic as a
doubly-filtered space to the corresponding knot lattice space for the dual knot
with the corresponding flip map. If we include the involutive data for the
original knot and ambient three-manifold, we can also produce the corresponding
involutive data on the new knot lattice space without assuming that the
original three-manifold is an $L$-space. I construct $\infty$-categories where
these operations are functorial. Finally, I use the surgery formula to compute
some examples of knot lattice spaces including for the regular fiber of
$\Sigma(2,3,7)$ and for a knot in a three-manifold that is not given by an
almost rational graph.
