  We study the fluctuations of the integrated density current across the origin
up to time $T$ in a lattice model of active particles with hard-core
interactions. This model is amenable to an exact description within a
fluctuating hydrodynamics framework. We focus on quenched initial conditions
for both the density and magnetization fields and derive expressions for the
cumulants of the density current, which can be matched with direct numerical
simulations of the microscopic lattice model. For the case of uniform initial
profiles, we show that the second cumulant of the integrated current displays
three regimes: an initial $\sqrt{T}$ rise with a coefficient given by the
symmetric simple exclusion process, a cross-over regime where the effects of
activity increase the fluctuations, and a large time $\sqrt{T}$ behavior with a
prefactor which depends on the initial conditions, the P\'eclet number and the
mean density of particles. Additionally, we study the limit of zero diffusion
where the fluctuations intriguingly exhibit a $T^2$ behavior at short times.
However, at large times, the fluctuations still grow as $\sqrt{T}$, with a
coefficient that can be calculated explicitly. For low densities, we show that
this coefficient can be expressed in terms of the effective diffusion constant
$D_{\text{eff}}$ for non-interacting active particles.
