While Dottore shows clear disdain for even the thought of actually being considered Anastasiy's father, I like to think that he's weirdly sensitive to his son's emotions, dubbed his "motherly instincts" by a teasing Columbina.
At first, it was nothing more than occasionally jerking out of a dead sleep because he felt like something was wrong. That something wrong being the child trying to hide his sniffles as he cuddled up on the couch in search of comfort- which Dottore can't have that, he'll catch a cold. The first few times it seemed to be a fluke, Anastasiy just happened to be upset and wandering around in search of a quiet place to cry where he wouldn't be in the way. But after a coincidental string of instances, Dottore starts to catch on to something being... off.
Anastasiy isn't connected to the 'hive mind' of the segments. It comes with his baby status, apparently, but Dottore swears up and down that he can feel something. Most of the Harbingers just brush it off as him losing his marbles again, but Anastasiy starts seeking him out more at night, seems to take comfort in him being nearby. Hell, sometimes Dottore will wake up first and end up hunched over the bed, half asleep by the time the kid wakes up in tears. Dottore isn't Irnes, he isn't comforting by a long shot, but the kid grows on him, and he finds himself curled up in a bed two sizes too small for him with Anastasiy more often than not. His back isn't exactly happy about the arrangement, but it's worth it.
The closer he is, the less Anastasiy seems to worry, and he finds that he was wrong. He remembers thinking his son was nothing more than a little bastard, a version of him that loved mischief and mayhem- a burden, really. But now... He's found that the kid has inched closer to him, occasionally showing him his drawings and giving Dottore books to read. There's no longer that sudden bolt of panic, just a warm feeling that Dottore has never felt before.
And when he feels the pain of injury, the sadness of being betrayed by someone so close yet so far, when his son is too big to hold in his arms...
He still goes to him. He tends to his wounds, holds him by his side, and ignores the distant pulling, telling him that something is wrong.
Dottore is not a good man. He abandoned Anastasiy when he needed him most. He let his own anger and frustration get in the way of Pantalone's attempts to be amiable for the child's sake. He's fucked, and he knows it.
But Anastasiy deserves better. And he'll be damned if he doesn't give him his best.