  We present a next-to-leading-order (NLO) study of the process $pp \; ( \to \{
h_1, h_2 \}) \to t\bar{t} + X$ in the 1-Higgs-singlet extension of the Standard
Model with an additional heavy Higgs boson $h_2$ that mixes with the light
Higgs boson $h_1$. This process is subject to large interference effects
between loop-induced Higgs-mediated amplitudes and the QCD continuum background
which tend to overcompensate any resonance contributions. A reliable modelling
of the resulting top-pair invariant mass shapes requires the inclusion of
higher-order QCD corrections, which are presented here. The computation of
these NLO corrections is exact in all contributions but in the class of
non-factorisable two-loop diagrams which are included in an approximate way
such that all infrared singular limits are preserved. We present numerical
results for several benchmark points with heavy Higgs masses in the range
$700$--$3000$ GeV considering the production of stable top quarks. We find that
the interference effects dominate the BSM signal yielding sharp dip structures
instead of resonance peaks. The significance and excludability of the BSM
effect is explored for the LHC Run 2, Run 3 and HL-LHC.
