It seems like a clever trick a foreign bank could pull but I'm not sure it would work.
So say 'B' acquires 1 billion dollars in base money from the USA ('A'). To import these dollars it had to send real wealth to the US (equipment, clothes, toys, services etc...) to buy these dollars initially. A heavy economic hit as they would be exporting real wealth and importing worthless pieces of paper intrinsically (dollars).
So does 'B' get its revenge when it turns one billion dollars in base money into say 20 billion dollars? I'm not sure... If somebody from country 'B' uses their account to purchase say 100k worth of farmland from an American, the transaction will clear reserves between the 'B' bank and the American farmer's bank in the states. So 'B' will be down more than 100k because it was multiplied and it would have to unleverage. If this was the case then the cost of adopting a foreign currency never outweighs the benefits.
But...if an American depositor trusts the the 'B' bank, then that changes everything and reserves don't have to leave B's banking system. In that case, yes country B did benefit.
So kind of a muddled answer, but it is hard to know if the cost or purchasing a foreign base money will be outweigh by foreigners interest in holding domestic bank accounts. You will always have conflicting pressures and it is tough to predict which will come out on top.
The scary situation for 'B' would be a confidence crisis. They don't have a central bank to bail them out. If investors get spooked and money gets pulled out of 'B' banking system there is nothing to stop a chain reaction and spectacular failure...which would happen absolutely...just a matter of when.
The moral of the story is the winner is whoever convinces everybody else that their "paper" or "1s and 0s" are worth something for real wealth. It could be base money in dollars, private money from American bank X, deposit money from say a French bank but denominated in American dollars or a central bank pegged currency. It all boils down to trust and faith in the end.