Maybe they believe Kyron is being held somewhere by an accomplice of TH. Maybe they are trying not to jeopardize Kyron's safety? I can't quite make those pieces fit, but it just crossed my mind. Probably not, but just maybe.
I'm sure they don't have enough to convict her. If they had enough to convict her, they would arrest her. Boy, I don't know, I'm just as clueless as everyone else.
I can't imagine at this point that Kaine does not believe that TH is in some way responsible for Kyron's disappearance. Either by direct act, or negligence. Otherwise, I agree, he would not split up their home so that Kyron could come back to a disaster.
Maybe LE suggested that Kaine get little KH out of there to try to gain some leverage on TH to get her to talk. As in
"If you want to see the baby again, you're going to have to spill some beans, Sister."
It's worth pointing out, that LE won't grant a restraining order to keep a mother away from her child without some significant evidence. That means Kaine has some reasonable supportable evidence to be able to request that he keep little sister KH and TH apart.