  Current observations of binary black-hole ({BBH}) merger events show support
for a feature in the primary BH-mass distribution at
$\sim\,35\,\mathrm{M}_{\odot}$, previously interpreted as a signature of
pulsational pair-instability (PPISN) supernovae. Such supernovae are expected
to map a wide range of pre-supernova carbon-oxygen (CO) core masses to a narrow
range of BH masses, producing a peak in the BH mass distribution. However,
recent numerical simulations place the mass location of this peak above
$50\,\mathrm{M}_{\odot}$. Motivated by uncertainties in the progenitor's
evolution and explosion mechanism, we explore how modifying the distribution of
BH masses resulting from PPISN affects the populations of gravitational-wave
(GW) and electromagnetic (EM) transients. To this end, we simulate populations
of isolated {BBH} systems and combine them with cosmic star-formation rates.
Our results are the first cosmological BBH-merger predictions made using the
\textsc{binary\_c} rapid population synthesis framework. We find that our
fiducial model does not match the observed GW peak. We can only explain the
$35\,\mathrm{M}_{\odot}$ peak with PPISNe by shifting the expected CO core-mass
range for PPISN downwards by $\sim{}15\,\mathrm{M}_{\odot}$. Apart from being
in tension with state-of-the art stellar models, we also find that this is
likely in tension with the observed rate of hydrogen-less super-luminous
supernovae. Conversely, shifting the mass range upward, based on recent stellar
models, leads to a predicted third peak in the BH mass function at
$\sim{}64\,\mathrm{M}_{\odot}$. Thus we conclude that the
$\sim{}35\,\mathrm{M}_{\odot}$ feature is unlikely to be related to PPISNe.
