This is quite an old question but I believe the answer to your question is given in Lemma 2.2 of Malle's paper "Height 0 characters of finite groups of Lie type" (2007) which is freely available online  <a href="http://www.ams.org/journals/ert/2007-11-09/S1088-4165-07-00312-3/">here</a>.

His lemma states that any two characters in the same (geometric) Lusztig series have the same central character. It follows very simply from the fact you gave for Deligne-Lusztig characters. This is because the characteristic functions of semisimple elements are explicit uniform functions, in the sense that they are linear combinations of Deligne-Lusztig characters.