  The X-ray emission of $\eta$ Carinae shows multiple features at various
spatial and temporal scales. The central constant emission (CCE) component is
centred on the binary and arises from spatial scales much smaller than the
bipolar Homunculus nebula, but likely larger than the central wind--wind
collision region between the stars as it does not vary over the $\sim$2-3 month
X-ray minimum when it can be observed. Using large-scale 3D smoothed particle
hydrodynamics (SPH) simulations, we model both the colliding-wind region
between the stars, and the region where the secondary wind collides with
primary wind ejected from the previous periastron passage. The simulations
extend out to one hundred semimajor axes and make two limiting assumptions
(strong coupling and no coupling) about the influence of the primary radiation
field on the secondary wind. We perform 3D radiative transfer calculations on
the SPH output to synthesize the X-ray emission, with the aim of reproducing
the CCE spectrum. For the preferred primary mass-loss rate
$\dot{M}_A\approx8.5\times10^{-4}$ M$_\odot$ yr$^{-1}$, the model spectra well
reproduce the observation as the strong- and no-coupling spectra bound the CCE
observation for longitude of periastron $\omega\approx252^\circ$, and
bound/converge on the observation for $\omega\approx90^\circ$. This suggests
that $\eta$ Carinae has moderate coupling between the primary radiation and
secondary wind, that both the region between the stars and the comoving
collision on the backside of the secondary generate the CCE, and that the CCE
cannot place constraints on the binary's line of sight. We also discuss
comparisons with common X-ray fitting parameters.
