That soon after the Dance, where House Lannister fought for the other side?
He’d probably see as giving the Lannisters too much power and influence compared to the Crown.
Would the Lannisters have needed his permission?
No snark, I seriously wonder. Tytos Lannister had extraordinary lending operations going on all the time; indeed, lending money at generous terms and then forgiving that debt was one of his big problems and one of Tywin’s first big political acts after the War of the Ninepenny Kings, with Tywin deciding “fuck this” and operating as a debt collector. The king didn’t need to be involved in any way.
This all involved mostly noblemen, of course, but I don’t see any reason these lending operations couldn’t be further extended. You’d just formalize it; establish the appropriate bureaucracy and infrastructure in Lannisport and let those classes and people that are capable of participating in borrowing, lending, and depositing know that the Golden Bank, backed by all the wealth of the Rock, is open for business.
Indeed, I would be surprised if there aren’t pseudo-banks already operating in Westeros. My understanding is that great merchant families and companies often operated as de facto banks; they’d issue debt, they’d bond and insure cargo, they’d take deposits, even sometimes have what were basically their own currencies. I imagine the Gulltown Arryns and the Lannisters of Lannisport have their fingers in such things.
I mean. You would, of course, want royal patronage and blessings, because that’s gonna make your life a lot easier and you can get the king to make laws to facilitate your business. But I’m unsure you’d have to go asking for that even to begin; Westerosi contract and finance law is pretty primitive but it seems to exist well enough to allow someone to just start a bank if they can convince people they’re trustworthy enough to operate as a banking .
For a formal bank, whose contracts and bills would be honored outside of the Westerlands, hell yes you need a royal charter.
And if you went ahead and did it without royal assent, that’s a quick way to find yourself accused of treason.