  We report observables for elastic Compton scattering from $^3$He in Chiral
Effective Field Theory with an explicit $\Delta(1232)$ degree of freedom
($\chi$EFT) for energies between 50 and 120 MeV. The $\gamma\,{}^3$He amplitude
is complete at N3LO, $\mathcal{O}(e^2\delta^3)$, and in general converges well
order by order. It includes the dominant pion-loop and two-body currents, as
well as the Delta excitation in the single-nucleon amplitude. Since the cross
section is two to three times that for deuterium and the spin of polarised
$^3$He is predominantly carried by its constituent neutron, elastic Compton
scattering promises information on both the scalar and spin polarisabilities of
the neutron. We study in detail the sensitivities of 4 observables to the
neutron polarisabilities: the cross section, the beam asymmetry and two double
asymmetries resulting from circularly polarised photons and a longitudinally or
transversely polarised target. Including the Delta enhances those asymmetries
from which neutron spin polarisabilities could be extracted. We also correct
previous, erroneous results at N2LO, i.e.~without an explicit Delta, and
compare to the same observables on proton, neutron and deuterium targets. An
interactive Mathematica notebook of our results is available from
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