I think it would make sense on either of the two, but the first one is plausible. The lady (not gonna mention her name cuz she not worth it) throws a fit by doing shitty decisions.
When she was married off to Yaetetsu, she dressed as an unmarried woman and had an affair to spite him and her family. Maybe she and her current husband had an argument about having kids. Out of anger of not getting what she wants again, she got away and left him, then went back to take O-ten.
And I agree, now that O-ten is five years old, she's less of a hassle and the lady thinks she can somehow manage take care of her.
I can relate to O-ten's side because I myself gotten to a similar situation, though my mother wants to be honest with me and told me everything when I was young. Same age as O-ten being 5 years old. At this point, now that O-ten has a bit of the truth to some extent, it would be best to be transparent with her than continue hiding. Hopefully, if O-ten gets to have the final say in this situation, she chooses to be with Yaetetsu and Toraji.
Yeah, I think O-ten is definitely getting the final choice. But she's somewhat prepared for it already with her limited knowledge. The important part would be for both of them to tell her though. Because it seems the mother wants to tell O-ten by herself, which is not good and really makes me think that she does not have good intentions. Also at the end of the day, she really only knows her Dada and those in the Inn. I don't think she would be prepared to just leave everything behind for a woman she barely knows anything about. It would take time, but the mother doesn't want that. I don't think she'll reject her mother outright, but she won't go with her.
I don't like the lady telling the truth alone with O-ten either, like you said, she might not have good intentions. I fear if she really does talk with O-ten alone, she could have the opportunity to twist some tales or even lie to her, who knows. And yes, I agree as well, even if O-ten would be able to be with her biological parents she wouldn't want to leave or not being able to see her Dada, Toraji, and the rest of the people at the Cheida-ya ever again. They were the ones who took real care of her and loved her dearly, not the lady who's basically a stranger to her.
I see one of two things going on: Her husband doesn't know what she's up to or they can't have kids.
It seems she's in a hurry, and based on the fact that she says that they are living proper lives, she and her husband may be working as traveling merchants. However, she seems desperate and somewhat sad, so it's not a situation where she JUST wants to reunite with her daughter. Based on the fact that she also dressed like an unwed woman and then ran away while abandoning her daughter, most likely to spite her ex-husband, she does not have a good track record of thinking through her actions. And her quickness to try and paint Yaetatsu as unfit means she's doing this on impulse.
So one theory is that she wants a family with her husband, but he doesn't want children. However now that O-ten is older, she's less of a hassle. So now the mother wants to take the chance that they are settled nearby to take her daughter and basically run away again like they did before, catching both her ex husband and current husband off guard.
The other theory is that the mother and her current husband want children but are unable to have any. This one is meh, but it's not impossible, especially because O-ten is their biological daughter and they can still raise her since she is five years old. This explains her insistence on Yaetatsu being a fake father, which she's crazy to think because he's the one who stepped up for O-ten. Yes, Yaetatsu is not perfect, and he's committed many mistakes but he's taken responsibility.
But let's appreciate how Yaetatsu would shoulder everything before but now he relies on Toraji (who is willing to do anything for his loved ones). I'm sure Toraji will be the one to mediate and get the truth out, because O-ten deserves to be raised in a loving environment with people she cares about. But also, she already knows the truth to some extent (she knows her mother left her). So O-ten is aware of more than anyone else thinks, and Toraji knows that as her new father figure.