As for a reason to go grab the Botany Bay: Brig. Having Khan on board is risky, he nearly took the ship and killed Kirk. He swings back, grabs the Botany, beams Khan and co over to the ship (after knocking out the engines) and impulse tows the ship back to Ceti Alpha V.
OTOH, depending on the astrography of the situation, going back for the Botany Bay
might merely add to the time Kirk has to keep Khan and his eighty-something cohorts in the brigs of the Enterprise
Generally speaking, giving a power-hungry villain his own hardware back is not a smart move. It's a bit like capturing a Klingon assassin and deciding his convict coveralls and cuffs aren't secure enough - and putting him back in his old clothes after removing the holster for the obvious sidearm. There's bound to be at least seventeen concealed weapons there!