I imagine Achamian and the Mandate have long since written off the Heron-Spear as lost. If they hadn't, they'd have spent time looking for it, since it's the only weapon known to be able to harm the No-God. It disappeared when the Scylvendi sacked the city it was in. If the Scylvendi Memorialists preserved the circumstances of their God's defeat, then it's likely the Scylvendi recognized the Heron Spear for what it was, and destroyed it to avenge their God. It's just a laser gun. It's quite destructible.
And even if it hadn't been destroyed, it's been two thousand years. Who can even say if its power source still works? It was the very last weapon of the Inchoroi. It could have finally failed.
As or Kellhus definitely having a plan to deal with the No-God...I'm not so sure that's the case. The entirety of his plan seems to hinge on getting to Golgotterath before the No-God is reborn. There would seem to be no other option, and no way to search for other weaknesses without capturing a very high-ranking member of the Consult who would actually have knowledge of what the No-God actually is. The Skin-Spies are not given knowledge like that. They're given precious little knowledge at all. Moenghus had been working on some for a decade, and had very little information to hare with Kellhus (though it's possible Kellhus had let Moenghus die before he could divulge all he knew).
Further, Cnaiur notes when talking with the Skin-Spies that their knowledge of themselves was curiously limited. They described themselves as servants of the Inverse Fire, but seemed incapable of saying what the Inverse Fire actually is.
I think this is part of the readership's consistent overestimation of Kellhus. The Great Ordeal is Kellhus' great gambit. This is his plan, and while I'm sure he has plenty of tricks in store, they're in service of the goal we see, not some hidden one.
Note that I don't blame anyone for overestimating Kellhus, mind. Almost all point of view characters, including Kellhus himself, view him as virtually infallible. The only exceptions are Moenghus and, much later, Maithanet. I believe Bakker is deliberately distilling a sense of Godhood about Kellhus, while also seeding the books with hints that said Godhood is false. They're small things. Kellhus doing something so simple as tripping in the chapter when he meets Moenghus, for example. But for the attentive reader, they're there. And I suspect we'll all marvel at the hints on re-read after we see what they're building too.