(1) the story hasn't ended. volume 8 is out but i don't think there are any translations yet. so we dk the baby's father and its very obviously not badma's since by his interaction with luka about him "just marrying the princess and having babies". also no sane baby daddy would be so enthusiastic hearing his brother is marrying the woman he knocked up.
(2) lala wouldn't want her child to be the head of wolsey's household. if im not mistaken she doesn't have a good relationship with wolsey because of what he did to ramdane. she wouldn't sacrifice her child just for ramdane to look like a paramore. the legitimacy of the child would be questioned as well. remember, the main reason why ramdane left for the 2nd time is because he didn't want to burden wolsey, who feels responsible for his mom's death and thus can't fight his dad, who isn't aware of their plot. wolsey did consider adopting a son from mcgrue but using the princess who's already pregnant would hide the kid's legitimacy (which i think is just for the sake of the public eye). the only problem now is if wolsey's dad actually cares about the legitimacy and if wolsey will ever tell him the truth.
The older brother's? I don’t get it. Why adopt a random baby when they could take in the baby of Wolsey's cousin who married Ramdane's twin sister?