We have control, draw and power to consider.
With Core+LOTR, you definitely want to be running The Power Behind the Throne, allowing you to engage in Intrigue challenges 2x per round. This lets you limit your opponent's options thru discard, and also to have a better chance of undefended challenges (since you can make 4 per turn, requiring more people to be knelt to deal with you)
LOTR Cersei is immediately essential, 3x, as she gives you power for Intrigue, doesn't kneel to attack Intrigue and picks up keywords as you win challenges.
Bronn's Hireling, 3x, because he is a cost 1/str1 Deadly Intrigue. Opponent's will be reticent about losing a character to "Deadly" on Intrigue, as they can't double-dip as with Martial, and may be more willing to simply discard a random card.
Doubting Septa, 3x, because she is a little bit of draw for a weenie who never hits the dead pile and will cycle into your deck as you do shuffle. She is there to die for claim or other purposes, though. Make no mistake.
Lord Tyrion's Host, 3x, will often be a cost 3 Str 6 bicon, as Lannister typically has gold "left over" / will not suffer much from having a bit set aside.
LOTR Myrcella, 3x, for sure - she gives you an additional intrigue challenge when you discard her, and is a respectable 2cost 2Str bicon when she's in play. Never dupe her - since her ability is a discard, you can pop it 3 times in one game.
Ser Ilyn Payne, 1x, because he's kind of awesome.
Even with only one copy, we will stick in a Core Jaime because he is great for military challenges and has martial/intrigue + deadly.
That is 17 cards, and we are really weak for military. Synergy-wise, we have gotten some good intrigue in, and from a keyword perspective have some ladies, clansmen, and knights with one queen.
So let's add in LOTR Bronn, 3x- he allows you to kill while defending by leveraging deadly. It's too bad we don't have access to House Payne Enforcers!
Now our gold curve is getting troublesome and our martial is still a big weakness. Let's add:
Tribes of the Vale, 3x. It means keeping your card count low, which can be hard when you're running Intrigue since you're milling through opponents, but we will focus this deck on having "playable" cards at all times to keep the count low.
LOTR Tyrion gives you some Stealth, combos up with your now (6) Clansmen, and is tricon+stealth. 3 of him.
This is 26 cards.
We will round out the Clansman theme with:
2x Crawn son of Calor, who lets you play Clansmen from Discard
1x Chella Daughter of Cheyk, who gives some mild location control.
We will add Core Pycelle, 1x, because he is great for draw.
That is 30 characters, but the troubling gold curve means we need some help from our locations.
Shadowblack Lane costs 0 and reduces an Intrigue play, so let's chuck that in. (non-limited) It's limited to one per deck.
Lannisport Treasury costs 2, but gives you 2 gold. If you don't get them out early there's not much point, but they're nice if you have some time to work with them. So we'll toss in 2.
Coin Mint is not great, but it is not limited and it is a net +2 coins, so let's add in 2 copies of that.
Golden Tooth mines, 1x, is mandatory for draw purposes.
Queen Cersei's Chambers, 1x, because it's cheap and non-limited.
The Gold Road, 3x, 0 cost. It's limited, but more flexibility than Treasury.
The Inn at the Crossroads, 2, just because it can be handy for you and plays into Clansmen.
That is 12 locations, all but one of which offer money or cost reduction. Let's add 3x Shield Islands Dromon for location control, and because we are going to be taking doing All The Gold in Casterly Rock as a plot.
30 Characters, 15 locations.
Now for attachments:
Increased Levy, 2x, can be played during setup, is unlimited, and can hook onto otherwise not-great lands such as the Treasury.
Bastard, 2x, is about all you're going to get for attachment control so learn to love it. You can use him defensively for your many non-power characters, remember.
Widow's Wail, 1x, can be nice when you draw into it as it boosts your draw., but as a unique isn't worth multiple copies due to ease of attachment control.
Bodyguard, 1x, why not? Some expensive uniques in here.
That's 6 attachments.
Terminal Schemes, x3, to kill pesky characters.
The Wealth of the Rock, x3, to boost income
Double Bluff, x3, to keep people guessing
That's 9 events.
That's 60 cards!
All the Gold in Casterly Rock for massive expenditures and emptying of hand to support clansmen (a bit… it is not really end of the world if you have cards in hand).
Frey Hospitality, for murder purposes.
Game of Thrones, for zerging.
Shadows and Spiders to limit opponent's options.
Mutual Enemies to paper over martial weakness.
Summoning Season to get Cersei or another key character out of your deck.
Valar or Wildfire for reset.
Use events are you get 'em, to bolster your board and to swap intrigue to power or martial for kill/power, or just to kill with terminal schemes. Play wildfire, and then follow it up with Frey Hospitality + Myrcella to clean up the board.
This is not a great deck, but it does have a story-ish theme (it's basically Cersei and Tyrion when he returns to King's Landing), should be able to win some, will struggle with accumulating/protecting power. Mulligan your first hand if it doesn't have 2 playable locations.