There are two types of people in the world:

1. Those who can extrapolate from incomplete data

Wasn’t there a guy who would actively remove artist signatures from comics because he considered it “advertising”?

Just hand wave away the “how,” if your story is about the person rather than about the laundering techniques. Just say that he has been laundering money and finally built up enough legitimate-seeming wealth that he can hold overseas.

Not every story needs to have an Andy Weir style explanation of the nuts and bolts of how each step progresses. Unless that’s what you want to do, in which case you better start consulting people who really know this stuff.

The amount of money the US has is, IIUC, slightly over $2 trillion.

About $5.5 trillion. Which, through banking, is multiplied as a money supply of about $23 trillion.

Honestly, my take is that you shouldn’t get into the weeds of foreign currency exchange, money supply, etc., in a fictional work unless you already have a strong grasp of the concepts involved. I personally don’t, just some macroeconomics coursework decades ago in college, but I know just enough to see when things don’t look right. If the plot allows for you to hand wave away how it works, you can just model things after existing banking havens throughout post-Bretton-Woods history.

When you deposit physical cash at a bank, the bank accepts the money, and records that in its books to show that your account has a higher balance. The account balance itself is your asset, and the bank’s liability. So when you have an account balance of $1000, that’s the bank owing you $1000, that they have to pay you at any time, on demand. They can pay it to you with physical currency, if you show up to the bank and say you want to withdraw the balance in cash. Or they can honor your demand to pay someone else on your behalf (through a paper check, electronic funds transfer, etc.).

Like if a bank takes the gym bag full of money, what do they do with the US$1 million

They can hold onto it, or deposit it somewhere else, in exchange for their own balance with some other financial institution.

The overwhelming majority of money in the modern system isn’t physical. It’s just records in a ledger: paper in the days of paper records, or digital records in the modern banking world. So although physical cash can move around, on the back end, the banks are mostly dealing with each other in updated ledgers, not physically moving physical currency. I might have a credit card account and a checking account with a bank, and I spend with my card in a way that increases the balance of the credit card, and then use my checking account to pay my credit card balance, and then all that gets resolved without any physical currency changing hands. Including with the merchant, who might accept credit card payments electronically, and then pay their employees through electronic payments as well.

Currency exchange is done by supply and demand. A small country with its own small economy and a floating currency is going to be vulnerable to swings in foreign exchange rates, so that country will likely need to hold a lot of reserves to shore up their own currency value, knowing that they have the foreign currency and gold to be able to trade it for their own currency, and keep the supply and demand for their own currency from completely swinging the value back and forth.

So if it’s a country that just uses dollars, and someone shows up with a bunch of dollars, their deposit helps shore up that bank’s own reserves of physical cash. Or if someone comes in with an electronic transfer of dollars to their account in the bank, that bank now has a higher balance of dollars on its records (recognized by the other financial institutions).

Adding a currency exchange component to this whole thing sounds like an unnecessarily complicated side quest, if you’re just trying to tell a story.

I suppose the country’s currency is hard, or reasonably hard, enough that it can be freely exchanged.

Why wouldn’t it just be another dominant world currency, like the dollar or the euro? In real life, banking havens in other countries still actually keep the accounts denominated in a currency that country doesn’t necessarily control: either pegged to a reserve currency or directly in that reserve currency itself.

The two fictional courtroom dramas that changed the course of my life (eventually persuading me I wanted to become a lawyer and make sure the system works towards justice) were A Few Good Men and To Kill a Mockingbird.

And even though recently things have seemed kinda bleak on the justice front in the big seats of power in the US (in the three branches of government), there’s still a million tiny ways to improve things locally, including in day to day interactions with others. So I still draw some inspiration from those ideals.