If I understand correctly, you're wanting the buyer to see the cost of their cart in your shop's default currency, and process the charge in your shops' default currency. You are not wanting to see the cart converted from Euro to your shops' currency. Is that correct?
I want the buyer to be able to pay in the currency of his choice.
The default shop currency is USD.
The buyer chooses EUR and puts the item in the cart.
The price in EUR is displayed in the cart.
Payment also in EUR.
The conversion takes place on the side of my bank in the account currency.
Thank you for confirming. You can use a currency switcher on your website to give your customers an idea of what they will pay before they process their order. The checkout can only support one currency, so the conversion will need to be handled by your customers' bank at the time of payment.