Catalysis of the reaction: 1,2-diacyl-sn-glycero-3-phospho-(1'-sn-glycerol) + L-lysyl-tRNA(Lys) = 1,2-diacyl-sn-glycero-3-phospho-1'-(3'-O-L-lysyl)-sn-glycerol + tRNA(Lys).
lysylphosphatidylglycerol synthase activity