Catalysis of the reaction: L-lysine + ATP + UDP-N-acetylmuramoyl-L-alanyl-D-glutamate = ADP + 2 H+ + phosphate + UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysine.
UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysine synthetase activity