I don't think anyone knew what they were going to do post-Dracula. Alucard planned to go back to sleep (he was most likely resigned to death) until Trevor told him to not make the Castle his grave but his new home and he bequeathed the Belmont Estate to Alucard as a way to give him a purpose.
Of course Alucard felt beholden to that and stayed.
Trevor expected Sypha to part ways with him and he never expected her to offer getting together. So obviously he didn't invite Alucard. On Sypha's behalf she probably thought Alucard liked being alone.
Pretty sure had he asked to join them they would have been OK with it, but both Trevor and Alucard were too proud/insecure to ask or it also didn't occur to them.