The Lady knows that she didn't tell the 'Fey Enchantress to do that, and the Changeling is in a world of hurt.
Quite possibly the atemporal Lady would know that there isn't a possible future where she would tell the Fey Enchantress to do so unless the Changeling is involved, in which case the Changeling is doubly screwed, as the Lady would know of the possible futures where the Changeling would take such a risk, and prepare a trap.
Messing with people under the direct orders of their gods would be dangerous for the Changeling.
He can walk into the throne room wearing the Fey Enchantress' shape to depose the King to find the Green Knight, half a dozen Grail Knights and an equal number of Prophetesses waiting with a tailored ritual to banish him for a long time.
Right, so like, The changling has cannoically gone into the literal doman of the Chaos God's diguised as their deamons and gotten away with messing around with them. Supposely cut the hair of Slanneh itself and ran away giggling, that's the degree upon which it can trick God's. It could go into the very heart of their power, and they didn't notice before it was too late.
And the Chaos God's themself certainly wouldn't display any sense of atemporal power where they could see into the future, since if they did they would have most certainlly retaliated quite forcefully against the Changling for daring to defy them, in the heart of their power. In which case i'm left to wonder how the Lady haven't united the entire world under her banner, if she's the solo God with the capabiltiy to see into the future like that, in cases where even the Chaos God's themself weren't powerfull enough.
In almost every instance, including this very quest, the Chaos's God's are the strongest, but your argument rest on the idea that the Lady has the ability to see into the future, be unaffected by the stream of time, by every instance that should mean there would be no way, Mousillon should fall, you'd have several failed Erran war's because the Lady would be able to see this coming in time, and diveret them for something else.
Even if you discount the possibility that the Lady could see into the future like that, it implies a sense of omniprescent that no other God's seem to quite have, since she would have to instantly discover that the Changling is disguised as the Fey Enchantress(Something it had gotten away with doing with all other God's, including the Al-Ulric highest of Ulric's servant if you take end-time into account). Then get into contact with every other possible party, get them to the castle in time, in order to stop it.
And even if she did manage to get the Green Knight and several Grail knights over there in time, She wouldn't know who is the real fey enchantress, since the Changling can trick even God's. They'd be stuck very much in a scenario of "i am the real one", and with no functional way of finding out.
None of the God's we have seen in the quests seem to have that sort of implied omnipresence and awarness, stuff like Nagash, Three Emperor's era, Chaos dwarfs would be radically different since the God's would know exactly when and where there is trouble brewing. Britania is just one example on how Changling could easily do something like that.
It could also just go "Heyo, it's your boy, the elector count of Nordland, and i've come to order my State troops to invade Laurelon, don't worry i got full permision, and this is totally within my rights as an elector count to do", Grombrindal "Kragg, you have dishonored every single one of your ancestor by helping that Umgi with your runes and sloppy craftsmanship" Miao Ying, "Open the Gates, We are letting out an party to cleanse the deamon lands, opps an enemy party was ready to just walk in"
If the only possible counter to the changling is to be an atemporal God who can see into the future, then you can surely see why the Changling might be what we could lightly call "gamebreaking and OP". From an story point, it only seems to not cause massive chaos and paranoia is because it's too busy causing mischief, and the fact GW isn't willing to let this creature into most stories due to it's inherent power