How to you intend to achieve that though? Tywin does not care about your orders and everyone else in the area can safely ignore you.
A difficult thing to do considering he has all the money, Prestige and troops
I mean we probably need to do it but how is sticky
By it being beneficial for them to do so. Even if they refuse the betrothal/marraige offers it's an honest attempt to reach out and make ties to cease the fighting. We've killed precisely zero starks. Meanwhile the Targaryens Mad king started Roberts Rebellion by killing the Lord of the North, his heir, and calling for the heads of Ned and Robert. The
Fyres are largely a bunch of Essosi sellswords, save of course for Euron Saltfyre, the Truest Targaryen to ever live. (also the maddest) And any Targaryen claimant has naught to offer the seven kingdoms but fire and blood.
We're offering weddings and the position of Hand, real offers, with the Weddings ensuring it that both our family and theirs have reason to cease the fighting and work together. Especially since the Targaryens and Stannis follow foreign gods with fondnesses for burning trees. All the Starks and Tully's have to do is say yes to our offer. We clear Edmure of the assassination charges, we heal the rift between them and the crown by marrying them to the Kings family and help head off any issues with their loyalists. Then, assuming both Accept the matches, or at least see it for the attempt at reconciliation and open some sort of dialogue we can join together the royal and tyrell armies,
and northerners and Riverlanders and use the four armies or the threat of such to call Tywin to kneel.
Then, we can use the combined forces to start seeing off challengers and mix our families blood with the Tyrells, Tully's and starks. Add in that Joffrey, Tommen and Myrcella each have Baratheon claims and we have most of the kingdoms Lords paramounts related to us by blood and working together. Even better if we have Shireen Baratheon also in hand.
Beyond that, having Edmure take his position as hand, and actually visibly be seen to be using it and ruling should go a long way to rebuilding trust after Ned was ousted if nothing else. And Like I originally argued for and was argued against, if nothing else we've got Cersei and Myrcella's hands to bargain with.
Perhaps an open display of force combined with an assassination?
Tywin will never cooperate with Cersei, and Tywin's very presence is undermining royal authority. He needs to be eliminated.
Assassinate our own father? No. Never.
And it certainly seems like he won't cooperate now. But if we can actually build a stable web of alliances. Such that him winning the war and keeping Joffrey on the throne simply by kneeling removes the Starks and Tully's as enemies and even turns them into allies... then he should accept. Or at the very least if he doesn't accept our rightful commands then with potentially four other armies set against him, to say nothing of any Lannisters who would be... confused at him not following the king. Then we'd have credible cause to quietly have him retired to casterly rock while another rules as Warden of the west and our Lord father enjoys a healthy retirement hunting and fishing in his old age.
Fun as all this is, Stannis is attacking King's Landing next season. Cearn Targaryen could also potentially break siege and go for us, though that would likely be disastrous. Regardless, holding the capitol is our is our predominant and current concern.
The ransom thing is done now. It didn't work, that's just how it is. Stannis is kind of insane, so I wouldn't expect hostages to ward him off anyway. This marriage plot that's been happening the last few turns really isn't how this all works. Marriages don't end wars, they cement pre-existing alliances. And yes, we have no authority to give Lannister betrothals to anyone so long as Tywin is still alive, even putting aside that Cersei would never send her daughter to the Starks.
Thus calling Joffrey to return to kings landing to seal the alliance with the Tyrells and for Edmure Tully to come take his seat as hand, hopefully and likely with an army. Then we can defend the city against any pretenders. Get marraiges wedded and bedded or at the very least betrothals signed. And once our position is secure use the armies to aid in clearing the crownlands of invaders.
And the ransom thing is most certainly
not done. We didn't pay the ransom. We wrote some letters promising a lordship over dragonstone if we pay the ransom
with safeies to ensure we actually get the hostages and the pirate doesn't just run off with the gold there's no reason to think we wouldn't get them. Ultimately, people tried to be clever and haggle instead of just paying the bloody asking price. It's refusing Renly's fucking pardon all over again.
As for authority to Marry off Lannisters. Power lies where men think it resides and where men want it to reside. Joffrey is king and Cersei the Queen Regent and eldest daughter of Tywin, heir to Casterly Rock. We absolutely have the Authority, if they want the Lannister matches then the Starks and Tully's can tie their cause with our own. And we also do have Myrcella's hand to bargain with for definite. Plus cersei's own. With four armies on our side if Tywin does not join our side then and hand over the Mountain he would have to be either the most stiff necked old goat to ever live or going senile in his old age. Either way, assuming he's not, he'd be all be certain to accept the alliances Cersei is trying to forge since it places a Lannister ass in almost every high seat in the realm.
We could always hurt Tywin where he feel the most: "the reputation of House Lannister". Make It seem that he has gone mad and paranoid in his old age, not listening to his grandson reasonable request to give up the custody of the Mountain - a man who was running around burning the lands of other Lord Paramount, is now a known kinslayer, rapist and other crimes - has activaly conspired against his daugther by bribing the Maester and is paranoid accusing Edmure Tully of all people of assasination.
Very bad idea. We're using Aerys madness as reason against Targaryen Pretenders. To try and claim our own grandsire is going mad chips away at our own reputation and credibility.
I wouldn't loudly call him out as insane. Just ask for his loyalty to Joffrey the king with the help of multiple armies. And
if he then refuses, we'd quietly sound out some other members of House Lannister and his household knights to help him step down and turn over his army to Joffrey Baratheon-Lannister the rightful king while Joffs beloved grandsire gets a break from the stresses of ruling over his long and distinguished career to enjoy some much needed leisure time.
Assuming Tywin doesn't yield when faced with multiple armies, which, he should, let's be honest.