  We compute the energy spectra of antideuterons and antihelium in cosmic rays
(CRs) in a scenario where hadronic interactions inside supernova remnants
(SNRs) can produce a diffusively-shock-accelerated "source component" of
secondary antinuclei. The key parameters that specify the SNR environment and
the interstellar CR transport are tightly constrained with the new measurements
provided by the AMS experiment on the B/C ratio and on the antiproton/proton
ratio. The best-fit models obtained from the two ratios are found to be
inconsistent with each other, as the antiproton/proton data require enhanced
secondary production. Thus, we derive conservative (i.e., B/C-driven) and
speculative (antiproton/proton-driven) upper limits to the SNR flux
contributions for the d and He spectra spectra in CRs, along with their
standard secondary component expected from CR collisions in the interstellar
gas. We find that the source component of antinuclei can be appreciable at
kinetic energies above a few ~10 GeV/n, but it is always sub-dominant below a
few GeV/n, that is the energy window where dark matter (DM) annihilation
signatures are expected to exceed the level of secondary production. We also
find that the total (standard + SNR) flux of secondary antinuclei is tightly
bounded by the data. Thus the presence of interaction processes inside SNRs
does not critically affect the total background for DM searches.
