I think it all rests on whether or not UnCat is interested in Jamie's lack of a sword hand. If she finds that maiming to be a fitting punishment for crippling Bran, than Jamie's will pass one huge hurdle, but still won't be out of the woods.
The second will be convincing UnCat that he personally was not involved in the Red Wedding and/or that he's more useful to her alive. If he mentions that Roose Bolton was the reason for his maiming, perhaps she'll forget about hurting Jamie at all.
I think he's too much like his brother: yes they are both flawed in good ways, but in the end, they are both insulated (by their wealth, abilities and position) from the real world.
Did he really believe that paying Bronn and Shae would make them loyal to him forever? Also, that's not his money he's using to buy them, it's Tywin's. Of course his father's angry that Tyrion brought a whore to court.
He claims to use his drawfness as an armor, but he still wants to be genuinely loved.
I've never been convinced he wasn't the real deal.
However, I believe Varys plans to bridge the divide between the Blackfyres and Targs by orchestrating them becoming one big family under one name.
The main reason there are no Targs save Dany left were all the dying Targs during Sir Duncan's time and the Summerhall disaster. If the Valerian descents can intermarry, it can only be a good thing--allowing for safer power shifts should madness show it's head again in a Targ ruler.
Varys would see to it and his position at court would be safe.
I'd say it's a mix of the Reach being on the wrong side of Robert's Rebellion, and because it's no secret that the Tyrells claim to Highgarden isn't the greatest and Mace Tyrell is an idiot. Also, since the Reach is wealthy in terms of food, the rest of Westeros is covering their options for when winter comes.