The J-matrix inverse scattering nucleon-nucleon potentials (JISP), describing both two-nucleon data and bound and resonant states of light nuclei to high accuracy, are tested on the total photoabsorption cross sections of H2, H3, and He3,4. The calculations in the three- and four-body systems are carried out via the Lorentz integral transform method and the hyperspherical harmonics (HH) technique. To this end the HH formalism has been adapted to accommodate nonlocal potentials. The cross sections calculated with the JISP are compared to those obtained with more traditional realistic interactions, which include two- and three-nucleon forces. Although the results of the two kinds of potential models do not differ significantly at lower energies, beyond the resonance peak they show fairly large discrepancies, which increase with the nuclear mass. We argue that these discrepancies may be due to a probably incorrect long-range behavior of the JISP, because the one pion exchange is not manifestly implemented there.