  We present 3D core-collapse supernova simulations of massive Pop-III
progenitor stars at the transition to the pulsational pair instability regime.
We simulate two progenitor models with initial masses of
$85\,\mathrm{M}_{\odot}$ and $100\,\mathrm{M}_\odot$ with the LS220, SFHo, and
SFHx equations of state. The $85\,\mathrm{M}_{\odot}$ progenitor experiences a
pair instability pulse coincident with core collapse, whereas the
$100\,\mathrm{M}_{\odot}$ progenitor has already gone through a sequence of
four pulses $1\mathord,500$ years before collapse in which it ejected its H and
He envelope. The $85\,\mathrm{M}_{\odot}$ models experience shock revival and
then delayed collapse to a black hole (BH) due to ongoing accretion within
hundreds of milliseconds. The diagnostic energy of the incipient explosion
reaches up to $2.7\times10^{51}\,\mathrm{erg}$ in the SFHx model. Due to the
high binding energy of the metal core, BH collapse by fallback is eventually
unavoidable, but partial mass ejection may be possible. The
$100\,\mathrm{M}_\odot$ models have not achieved shock revival or undergone BH
collapse by the end of the simulation. All models exhibit relatively strong
gravitational-wave emission both in the high-frequency g-mode emission band and
at low frequencies. The SFHx and SFHo models show clear emission from the
standing accretion shock instability. For our models, we estimate maximum
detection distances of up to $\mathord{\sim}46\,\mathrm{kpc}$ with LIGO and
$\mathord{\sim} 850\,\mathrm{kpc}$ with Cosmic Explorer.
