Superlattices (SLs) comprising layers of a soft ferromagnetic metal La_2/3Sr_1/3MnO₃ (LSMO) with in-plane (IP) magnetic easy axis and a hard ferromagnetic insulator La₂MnCoO₆ (LMCO, out-of-plane anisotropy) were grown on SrTiO₃ (100)(STO) substrates by a metalorganic aerosol deposition technique. Exchange spring magnetic (ESM) behavior between LSMO and LMCO, manifested by a spin reorientation transition of the LSMO layers towards perpendicular magnetic anisotropy below T_SR = 260 K, was observed. Further, 3ω measurements of the [(LMCO)₉/(LSMO)₉]₁₁/STO(100) superlattices revealed extremely low values of the cross-plane thermal conductivity κ(300 K) = 0.32 Wm⁻¹K⁻¹. Additionally, the thermal conductivity shows a peculiar dependence on the applied IP magnetic field, either decreasing or increasing in accordance with the magnetic disorder induced by ESM. Furthermore, both positive and negative magnetoresistance were observed in the SL in the respective temperature regions due to the formation of 90°-Néel domain walls within the ESM, when applying IP magnetic fields. The results are discussed in the framework of electronic contribution to thermal conductivity originating from the LSMO layers.