Interact? That shouldn't have played out, because he had Elza steal the DB for that very reason, and I don't see why she couldn't have stolen the CK AK at the same time.

What it more plausible, though, is that Shaidar Haran had no idea the CK AK was there in Cadsuane's stuff, and he ordered Elza to steal the DB, and said nothing about the AK - so in her terror of SH she took no initiative and left the AK there and said nothing about the fact it was there (if we look at Alviarin's reactions to SH, Elza probably didn't dare say anything but "yes, great lord" to SH.)

Another possibility is that Shaidar Haran can't feel anything special about ter'angreal/angreal/san'angreal - cannot distinguish them from the rest of the pattern/reality (which is also woven of saidin/saidar - everything in Creation is), so just by looking at or touching the object he could't tell this in "a OP object". He may have overlooked the CK AK because he never cared to ask a Chosen (or Elza, while she undid the ward following his instructions) what it looked like and didn't recognize it in Cadsuane's trunk. In other words, SH might have ordered Elza to steal the CK AK had he known what this statuette was, but he didn't know what it was and couldn't care less to ask, since what he wanted was the DB, not Cadsuane's baubles). A human being would have thought to wonder what could be hidden under a ward like this, but RJ made a point of repeating that Shai'tan doesn't think on human terms at all.

What I think however is that Shai'tan left the CK AK there on purpose. It seems obvious that Moridin's recent strategy for Rand (and thus, most likely, Shai'tan's) was to make him desperate and destructive and darker and darker - and more and more the CK would become an asset, something with which Rand could cause much greater harm to the Pattern than to Shai'tan. Moridin all but told Rand "please go on a balefire rampage to kill all the Chosen" (the key concepts being: overlook the dangers of balefire, and shed away your mental block about killing women in cold blood.. not a coincidence Moridin sent two women against Rand recently), and he has changed his approach to make Rand do what he wanted: it's no longer "you will be rewarded beyond measure, with riches, power and immortality if you join me" - Moridin knows this works no more with Rand than it did with LTT - it's more like "Don't you agree Creation sucks and putting an end to it all would bring us the rest we both want. I'm not lying anymore, I don't want power and I won't offer you power. We're alike both of us - these lies are good for the fool Chosen, what we both want is to rest, for all the pains to end for good". It's more "either you join us and Shai'tan will give you the final rest you want, or destroy Creation yourself if in your stubborness you won't join Shai'tan - the end result will be the same". I think Shai'tan wanted Semirhage to capture Rand and bring him to SG in order to put him in a situation of such grain pain and despair his only way out would be to embrace the TP to put an end to that. He would have released "dark Rand", either turned for good, mindtrapped or just made dark by an addiction to the TP (the last might be better, turning to the Shadow doesn't make you loyal, just "darker" - by the way I think this is what caused Taim's "dark aura", ie: he's been turned against his will). Semirhage unwittingly created this situation of pain and despair where Rand might use the TP by forcing Rand to torture Min, and so Shai'tan opened access to Rand, Rand took the bait and so Shai'tan had what he wanted and didn't need Rand captured and brought to SG anymore (it would be like Shai'tan and the Shadow's general arrogance to believe a taste of the ecastasy of the TP was enough, and Rand would be his for good soon). He let Rand kill Semirhage, because what Shai'tan gained by the fact Rand has embraced the TP willingly was worth far more than Semirhage to him at this point.

I think the Shadow's biggest mistake was to have overlooked the CK. It seems to me, at least, that Rand was able to resist the temptation of embracing the TP again because he had the CK to embrace a massive quantity of saidin if he felt the need (it seemed to feel like embracing a much vaster amount of power than Rand actually embraced, ie: the ectasy threshold with the TP seems much lower than the OP's, which takes much greater amounts to produce the same exihilaration. It takes a lot less TP to feel the way you would feel using the CK or even Callandor. In any case, it's after touching the TP that Rand started wanting to have the CK with him at all time, and I think there's a connection. That sounds like a typical drug addict inconscious reaction - he wanted the assurance that he could get "his fix" whenever he needed it. If we compare that to the AS fears about using too much saidar too often and becoming more and more unable to resist using big quantities often, it might mean a channeller with no huge san'angreal/no possibility to join big circles who tasted the TP might rapidly have start to feel using saidin "underwhelming" and seek to touch the TP again, and the more he does that, the more addicted he becomes, and eventually he gives up saidin altogether. How fast is probably a matter of will, and predispositions. Some people can resist addiction better/longer than others.

So.. it seems Rand was first "saved" by the fact he knew the TP was Shai'tan and evil, and he had the CK to compensate, so he didn't feel compelled to use the TP again. It means Shai'tan failed because SH has overlooked the CK in Cadsuane's things.

I get the feeling now he doesn't have the CK anymore, and Callandor is for the moment unaccessible, Rand might begin to have to fight the temptation of embracing the TP.