  We introduce the transition-density formalism, an efficient and general
method for calculating the interaction of external probes with light nuclei.
One- and two-body transition densities that encode the nuclear structure of the
target are evaluated once and stored. They are then convoluted with an
interaction kernel to produce amplitudes, and hence observables. By choosing
different kernels, the same densities can be used for any reaction in which a
probe interacts perturbatively with the target. The method therefore exploits
the factorisation between nuclear structure and interaction kernel that occurs
in such processes. We study in detail the convergence in the number of partial
waves for matrix elements relevant in elastic Compton scattering on $^3$He. The
results are fully consistent with our previous calculations in Chiral Effective
Field Theory. But the new approach is markedly more computationally efficient,
which facilitates the inclusion of more partial-wave channels in the
calculation. We also discuss the usefulness of the transition-density method
for other nuclei and reactions. Calculations of elastic Compton scattering on
heavier targets like $^4$He are straightforward extensions of this study, since
the same interaction kernels are used. And the generality of the formalism
means that our $^3$He densities can be used to evaluate any $^3$He
elastic-scattering observable with contributions from one- and two-body
operators. They are available at https://datapub.fz-juelich.de/anogga.
