I just have to add to this thread, because for me, this was the stupidest the show has ever been, and the problems with this plan go far beyond whether or not Cercei will believe them.
Let's recap - why have our heroes decided on this plan to go and capture a Wight and return with it to King's Landing? Because apparently Jon is informed that the Night's King is marching towards EastWatch, and there's no time to wait until the war is done to go and fight him, they need Dany's help to fight him now. But Dany cannot abandon the war, and needs to have an armistice with Cercei first, and convince her that the threat is real.
So given that we are incredibly pressured for time, what is the plan that our heroes come up with?
Well first of all, they need to have a council to discuss the news. Remember, Dany flies back, but Tyrion cannot do so, so obviously we need to wait for him to sail back from King's Landing to Dragonstone in order to have the meeting.
Then after we've made the decision to rush back and capture the Wight we need to... wait for Davos to smuggle Tyrion back to King's Landing to have a 10 minute conversation with his brother, get Gendry, then sail back to Dragonstone.
At which point, Jon, who is King in the North and lords over thousands of men, will sail on his own with an untrained blacksmith, an old man, and Jorah Mormont, on to Eastwatch. There, together with another 4 men, they will walk out north of the wall, with no horses and no supplies, against an army of tens of thousands of dead zombies, that they have no idea what place they are in, and where they will somehow capture one of the zombies (how? what it the plan to keep the thing immobile?). After they capture the zombie, the seven of them plan to walk back to EastWatch.
By the way, remember at the end of Season 1 where an expedition North of the Wall was a big deal that involved most of the manpower of the Night's Watch, and took about 3 seasons worth for the entire plot to be resolved?
Then after they've captured this zombie, and assuming our 7 heroes haven't been killed by the ravenous army of a hundred thousand zombies, they will sail back from EastWatch to King's Landing, Dany (who's apparently just sitting around doing nothing for all the months if not years this whole plan will take) will sail back from Dragonstone, they will have a meeting with Cercei where they will show her the zombie. Then if things turn out well, Dany will march back to the North, where apparently the Night's King is still waiting before attacking East Watch.
This plan, which involve at least two treks along the entire length of Westeros, multiple trips from Dragonstone to King's Landing and back, and an expedition into the unknown beyond the wall, is what the brilliant and cunnint Tyrion came up with in order to save time given how urgent the situation is.
Here's what bothers me - I don't mind dragons, and magic, and the fantastical elements, given that the plot makes sense in the context of the fantastical elements established. What bothers me is, given the context of the setting, when characters do actions that make no sense given that context, and which in reality they never would. And nobody in the context of the story would actually come up with this plan.
I remember watching the commentaries for season 4, at the scene where Brienne fights the Hound, and I'm sure I remember D&D were commenting that part of how they decide how to deviate from the book, is when they imagine what would happen if two characters who are separated in the books were to meet. So the fight with Brienne and the Hound happened because they felt 'Wouldn't it be cool if we could get to see these two characters fight and find out who would win?
So what annoys me about this episode, is that clearly, the plot is not being set up to have internal consistency with the world, or the characters, but to answer fanboyish 'What if' questions. D&D think it would be cool if the Hound and Jorah Mormont met together and fought a White Walker, so by God, they'll find a way to make it so. The show is constructed to think up of 'cool' set-pieces, and then things are made up to make those set pieces possible, no matter how non-sensical given the context of the characters or the setting.
Jorah Mormont, who is in love with Dany, and has just survived greyscale and managed to return to her, would never just abandon her to go off onto an expedition north of the wall. But the cool set-piece demands that he do so, so he will.
Jon Snow, King in the North, and ex-commander of the Night's Watch, would never go on an expedition North of the Wall with 7 men and no supplies. But the cool set-piece demands that he do so, so he will.
Remember that the Brotherhood without Banners is a 'brotherhood' of probably tens if not hundreds of men? But everybody in the brotherhood disappears into thin air without explanation, with miraculously only the 3 named members remaining, who just so happen to be captured by Tormund, just at the time that Jon happens to want to go North.
It's all so... stupid. What made Game of Thrones interesting was the complex 3-dimensional characters, and the de-construction of fantasy tropes given the brutal reality of the world. Remember for example in Season 1, when Jorah Mormont fights the bloodrider, and defeats him because of the protection afforded to him by his armor? Despite the typical fantasy cliché of the highly skilled steppe warrior, we are shown how in real life, the practicalities of armor and medieval combat are effective in a real fight. Now we get the cringe-worthy line from Jaime about how 'Now I've seen the Dothraki fight, no army in the seven kingdoms can defeat them' (Why?). But now everything, from the Dothraki to the individual characters, has been devolved to one-dimensional fantasy clichés that GRRM tried so hard to elevate his books from, and the plot has devolved to a series of nonsensical actions to go from one action set-piece to another.