The Enterprise warps back to the Botany Bay, takes everything to CA5, settles them on the planet by bringing (somehow) the cargobays from the BB to the surface, which could have taken a while, got the colonists down there, Spock likely scanned the region while they were there, then left.
To nitpick, the props and sets tell us this did not happen, not quite this way.
The huts down on the hellhole planet are not
Botany Bay cargo elements - they are Starfleet containers, with Federation logos all over them. So we don't have to assume Kirk would have gone back to the derelict spacecraft. Rather, we could assume the simpler thing: that Khan, when beaming over his 72 compatriots for his attempt at taking over the starship, also brought along his personal belongings.
It was Khan who cut loose the ancient spacecraft and proceeded with the starship only, not Kirk, so Khan would no doubt have arranged for the transfer of essentials. Not colonization essentials, but starship takeover essentials, such as his book collection, with a genuine
Botany Bay collectible strap around them. But Kirk wouldn't have to rely on Khan for colonization gear: his own ship would be amply stacked with it, and he and he only would decide how it gets expended and how this expenditure gets written down in the records. And indeed none of the stuff we see down on the huts is explicitly 1990s colonization gear.
As for Carol Marcus knowing that the torpedoes have warheads, well, there's no reason why she should. She may be aware of the specs, but the alterations would not be in the specs. And beyond that, she has merely looked at the things from the outside, and Scotty confirms this doesn't tell anybody anything.
Whether Alexander Marcus knew... He says half a dozen things to Kirk that are unlikely to be true, simply because he's a hardened liar who would win nothing by telling the truth but hopes to score points with every lie. Is Khan a "war criminal", say? Highly doubtful, as claiming so would be Marcus' best hope for breaking off the dangerous Kirk/Khan relationship. What would Marcus win by sending the 72 sleepers out into space, though? His plans gain nothing from this, while letting Khan believe the sleepers are there but in fact placing them in a vault back on Earth again would work to Marcus' advantage.
Now imagine carpet bombing somewhere on Earth with 200 nuclear bombs. That would be a massive area of destruction causing an insane nuclear winter if not just completely obliterating life on the planet for a long time. That's the definition of overkill if your target is just one man.
What's the alternative? Using just one nuke would probably not kill Khan, because they have no idea where in the vast desert he might be hiding - the sensors don't reach that far out, supposedly, so it would be up to terminal homing, and essentially chance. OTOH, the downside of using 72 nukes is zero: the Klingons will go berserk from even one photon grenade anyway, but 72 nukes won't result in any greater collateral damage than 1, not in that wasteland.
In pure tactical terms, then, nuking the Klingon homeworld with 72 warheads is a smart move for securing a kill and not having to spend extra minutes in the danger zone doing damage assessment. This ain't going to be "surgical" no matter what Kirk does.
Timo Saloniemi