This work is concerned with transmutation of the highly toxic and radioactive isotopes Cs-135 and Cs-137, which are extremely difficult to transmute in the conventional neutron fields, together with I-129. In this study, a novel strategy of photon and neutron hybrid transmutation has been proposed for transmuting a CsI target. This process uses high intensity and high energy gamma rays for the giant dipole resonance (GDR) based photonuclear transmutation and then utilizes the emitted neutrons for further neutron capture based transmutation. From a comparative study of photon sources, the laser-Compton scattering (LCS) gamma ray source with monochromatic nature and energy tunability is chosen to pursue the hybrid transmutation. The spectral optimization of the LCS gamma ray source has also been performed to enhance the photonuclear transmutation and reduce the unnecessary heating due to photo-atomic reactions. In this research, the hybrid transmutation process is simulated by directing the optimized LCS gamma rays towards a central CsI target for photonuclear ($\gamma$,n) reactions. Energetic neutrons resulting from the GDR ($\gamma$,n) reactions are scattered through a moderator into surrounding hexagonal CsI blanket targets where they undergo both moderation and neutron capture (n,$\gamma$) transmutation reactions. The hexagonal geometry is chosen as it can minimize the leakage of neutrons from the blanket region. Design parameters such as target dimensions and lattice pitch size have been optimized to enhance the effectiveness of the transmutation. Both light and heavy water are compared as the neutron moderators to maximize the neutron-based transmutation rates. An innovative rotating central CsI target has been proposed to disperse the heat deposited by a focused LCS beam source. Furthermore, vertical and annular heat conduction pathways (HCPs) have been introduced and optimized to enhance the overall thermal conductivity of the central CsI target. The rotating CsI target in combination with the HCPs enables effective removal of heat from the central target bombarded by a high intensity LCS beam. Based on the transmutation and thermal analyses results, it has been concluded that the hybrid transmutation has promising potential for the radioactive waste management.