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 historys 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. Largest scatter is achieved by growing late ($zleq1$) major mergers that spread out existing stars to create massive, in-situ dominated stellar haloes. Increasing a last major merger at $zsim2$ 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.
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