^62,64Cu are radioisotopes of medical interest that can be used for positron emission tomography (PET) imaging. Moreover, ⁶⁴Cu has β^? decay characteristics that allow for targeted radiotherapy of cancer. In the present work, a novel approach to experimentally demonstrate the production of ^62,64Cu isotopes from photonuclear reactions is proposed in which large-current laser-based electron (e^?) beams are generated from the interaction between sub-petawatt laser pulses and near-critical-density plasmas. According to simulations, at a laser intensity of 3.4 × 10²¹ W/cm², a dense e^? beam with a total charge of 100 nC can be produced, and this in turn produces bremsstrahlung radiation of the order of 10¹⁰ photons per laser shot, in the region of the giant dipole resonance. The bremsstrahlung radiation is guided to a natural Cu target, triggering photonuclear reactions to produce the medical isotopes ^62,64Cu. An optimal target geometry is employed to maximize the photoneutron yield, and ^62,64Cu with appropriate activities of 0.18 GBq and 0.06 GBq are obtained for irradiation times equal to their respective half-lives multiplied by three. The detection of the characteristic energy for the nuclear transitions of ^{62, 64}Cu is also studied. The results of our calculations support the prospect of producing PET isotopes with gigabecquerel-level activity (equivalent to the required patient dose) using upcoming high-intensity laser facilities.