Spontaneous and photo-induced decay processes of HfF₅⁻ and WF₅⁻ molecular anions were investigated in the Double ElectroStatic Ion Ring ExpEriment (DESIREE). The observation of these reactions over long time scales (several tens of ms) was possible due to the cryogenic temperatures (13 K) and the extremely low residual gas pressure (∼10⁻¹⁴ mbar) of DESIREE. For photo-induced reactions, laser wavelengths in the range 240 to 450 nm were employed. Both anion species were found to undergo spontaneous decay via electron detachment or fragmentation. After some ms, radiative cooling processes were observed to lower the probability for further decay through these processes. Photo-induced reactions indicate the existence of an energy threshold for WF₅⁻ anions at about 3.5 eV, above which the neutralization yield increases strongly. By contrast, HfF₅⁻ ions exhibit essentially no enhanced production of neutrals upon photon interaction, even for the highest photon energy used in this experiment (∼5.2 eV). This suppression will be highly beneficial for the efficient detection, in accelerator mass spectrometry, of the extremely rare isotope ¹⁸²Hf using the ¹⁸²HfF₅⁻ anion while effectively reducing the interfering stable isobar ¹⁸²W in the analyte ion ¹⁸²WF₅⁻. The radionuclide ¹⁸²Hf is of great relevance in astrophysical environments as it constitutes a potential candidate to study the events of nucleosynthesis that may have taken place in the vicinity of the solar system several million years ago.