Wilhelm and his father ordered Rein to bring Billy back to the imperial family because he has imperial blood. She had no choice. Rein tried raising Billy, she even breastfed him at first, but she couldn't stand to look at him as he got older and started to remind her of Wilhelm. She didn't abandon him at this point though, she passed him off to her dearest friend to raise. Earlier on we saw that she wanted to give the kid a chance because her own stance is that he shouldn't be held responsible for her hatred of Wilhelm. Wilhelm had been lying to her and manipulating her and she was already pregnant by the time she found out what he did. So for her he was basically a rape child that she didn't want to abort, but couldn't bond with. She had post partum.
The above is why Rein left Billy with the imperial family, why he can't be raised by Dietrich and why Dietrich is doing what he is now. He raised the kid and feels for him. He feels bad for Billy, who he's been lying to saying Rein has been too busy to visit while Wilhelm has spent the last like two years completely neglecting Billy, not even checking in on him. He's seen him at like two social events he was required to attend. He's asking Wilhelm to bring Billy to see his mother because it's what Billy wants and because he sees that things aren't working as they are.
Wilhelm is the problem currently, Rein just chose to birth his child and at least make sure the child is being taken care of by someone good till she was forced to return him by his own plan to get her to him so he could try to convince her again, only to rape her for three days straight. Though this recent chapter seems to imply a change of character in him to becoming "emotionless" from his previous "mad(insane) and wrathful", not sure if that's positive or not. I'm guessing it's to segwey into the spoilers people have discussed here with them ending up together again, which could only happen if he cuts his manipulation and controlling ways.
This is a good analysis of the plot's progression
Now I'm starting to hate Dietrich, can't they just leave Wilhelm alone? It's annoying, why should he go to the north for the prince to see his mother when his own mother doesn't even want to see him