  We introduce and apply a new approach to probe the response of galactic
stellar haloes to the interplay between cosmological merger histories and
galaxy formation physics. We perform dark matter-only, zoomed simulations of
two Milky Way-mass hosts and make targeted, controlled changes to their
cosmological histories using the genetic modification technique. Populating
each history's stellar halo with a semi-empirical, particle tagging approach
then enables a controlled study, with all instances converging to the same
large-scale structure, dynamical and stellar mass at $z=0$ as their reference.
These related merger scenarios alone generate an extended spread in stellar
halo mass fractions (1.5 dex) comparable to the observed population, with the
largest scatter achieved by growing late ($z\leq1$) major mergers that spread
out existing stars to create massive, in-situ dominated stellar haloes.
Increasing a last major merger at $z\sim2$ brings more accreted stars into the
inner regions, resulting in smaller scatter in the outskirts which are
predominantly built by subsequent minor events. Exploiting the flexibility of
our semi-empirical approach, we show that the diversity of stellar halo masses
across scenarios is reduced by allowing shallower slopes in the stellar
mass--halo mass relation for dwarf galaxies, while it remains conserved when
central stars are born with hotter kinematics across cosmic time. The
merger-dependent diversity of stellar haloes thus responds distinctly to
assumptions in modelling the central and dwarf galaxies respectively, opening
exciting prospects to constrain star formation and feedback at different
galactic mass-scales with the coming generation of deep, photometric
observatories.
