I am not sure the story change different in light novel.
In my opinion. This not anastachius, this carax or illegitimate child from first emperor (he really fans lucas) but he can't be king cause he didn't have strong magic like everyblood noble king (he meet lucas when the first emperor introduce him), lucas really don't like him cause he mixed ( I think he make a black magic and become magician so he can long life like lucas).
yes I believe he was the son he rejected
https://who-made-me-a-princess.fandom.com/wiki/Aeternitas_de_Alger_Obelia

Someone please correct me if I’m wrong, isn’t Anastacius (?) the guy who Lucas is/was said to be closed to?? Like, in the early part of the story, there was an emperor Lucas was friends with. ALSO, MY HEART IS WEEPING!!! DID YOU SEE THE CREDIT PART?! WHY CAN’T WE HAVE A WHOLE FAMILY T^T