Reserves aren’t necessarily in the form of coins – they can also in the form of bullion, deeds to real estate, stocks and bonds, etc. So the issue is that the Golden Bank would be circulating notes backed by gold bullion, which represent an alternative currency. And in a world that runs on metallic standards, a lot of people are going to be happy to take Casterly Rock’s gold.
It’s going to cause institutional fracture regardless, because what’s at stake is who controls how much money is in circulation. Even if the reserves were in golden dragons, who decides how many notes the Bank can circulate per dragon (i.e, what the bank’s reserve rate is)? Who decides how many dragons get minted per year? You’re going to get fights between the Bank and the Crown for authority.