Catalysis of the reaction: 1D-myo-inositol 1,3,4,6-tetrakisphosphate + ATP = 1D-myo-inositol 1,3,4,5,6-pentakisphosphate + ADP + H+.
ATP:1D-myo-inositol-1,3,4,6-tetrakisphosphate 5-phosphotransferase activity