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 here.
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.
