An additional point that hasn't been addressed: it's unlikely Kirk had any authority over Reliant's
assignment to the Genesis project because if he had, it seems to me he'd have contacted Reliant
after getting Carol's message rather than contacting Starfleet Command. But not only did he not do that, he also didn't seem to make the connection between Genesis and Reliant's
sudden appearance in the Enterprise's
path hours later. It's only after Khan asks about Genesis that he realizes something's up.