It took me far too long to realise that the reason Quaritch kidnapped Spider before confronting Jake and placed him in Bridgehead, is because Spider is his weakness and he knows it.
Miles’s A3 storyline focuses on the consequences of his reckless actions in A2, and you can feel the collar tightening around his neck. He has lost status and is in the worst position we have seen him so far; a dog on a tight leash, being threatened to get put down if it bites the hand that feeds again.
Quaritch is careful about outwardly showing it, but he is not in denial about his feelings towards Spider. He loves him and, despite Socorro’s burning hatred for his father, deep down, that love is mutual (canon info taken from Pandora exploration book). Miles also knows that Spider can and will be used against him if he confronts Jake in the open — it happened before, and time and time again the mission got compromised because of his parental duty.
So Quaritch did the logical thing and removed Spider from the playing board. He had to ensure he wouldn’t get in the way or be dangled in front of him like a carrot, but because Miles pushed the air mutation angle to get RDA to agree to pursue him, he is thereby forced to place him in a scy-ops facility.
I hated him for that and to be clear, I still do, but at that point in the story, it wasn’t something Quaritch had any control over. He ran himself into a corner and the only way he could keep Spider was on RDA’s terms. He had no say on his conditions or his destiny at that point, but he took the risk because having Socorro contained and nearby meant that he wasn’t going to compromise Miles again by twisting the knife buried deep in his rotten heart.
Recombinant Quaritch is a horrible person for this, but not as horrible as some people say when they assume he deliberately chose to place him into a scientific research facility, as opposed to just being a short-sighted idiot who doomed his son to a dangerous fate.
Basically — Quaritch kidnapped his son because he cares for him deeply and knew the danger that bond posed to him. He knows he loses control over his thoughts and emotions whenever his son is in danger or on the opposing side, so he had him taken to Bridgehead where he could be close, but not threatening to Miles.
However, Quaritch had little to no power left when Spider was finally brought in, so he had to become okay with his sci-ops conditions, because at least he was close by.