You have a fundamental economic mistake going on here:
"The coins contain precious metals and weigh too much to carry more than a few, and the banknotes, while lighter, are big and unwieldy and don't handle folding too well."
You are basically mixing gold-backed currency with government-backed currency. Today we have government-backed currency. Which means that none of our currency - even the penny - is worth in precious metal value what it's face value us. But you are saying the coinage is "precious metals" that is the actual metal of the coin currency has value - while the banknote value of the paper the banknote is on is zero.
For currency to exist that has no actual value of the currency itself, you have to have a government around that will back it. This is why coins were used in ancient times - because for example the government of one city would have given zero value to the government of another city. You could, for example, take a wagonload of wheat to 1 city where it was worth maybe 1 gold coin, then take that coin to another city and buy maybe 2 wagonloads of wheat (because in that other city, wheat was more common) and then spend time going back and forth transporting wheat and amassing a large stack of gold coins. 130
(and there were people that did just that)
But a scheme like that could not exist if you took a banknote in trade for the wagonload of wheat because when you got to the other city, the banknote would be worthless.
You have to either completely dispense with government-backed currency (which means coins and gems only) or dispense with gold-backed currency (coins where the value of the metal is equal to the value of the coin). If you mix the two you will have people melting down coins for the metal and thus destroying the economic system because they will be affecting the money supply.
If you go with gold backed currency then the fairies can carry precious gems pretty easily (small ones, obviously)
If you go with government-backed currency then that solves the problem because the "fairy government" will back whatever tiny banknote is printed that is easy for the fairies to carry so the merchant will happily accept it because at the end of the week he can take the thimble full of fairy banknotes to the fairy government and get it exchanged for "big people" banknotes.
In Fantasy stories people often like to dispense with the simple fact that currency is mainly tied to governments. Governments are annoying things to have around since they introduce politics and politics complicates a simplistic idea of Good and Evil. So you have lines like from Lord of the Rings where <PERSON> compensates for stolen horses with "12 silver pennies" with zero mention of what government minted these pennies, where the silver came from, whether the pennies will be even of any value the moment they leave Bree, or if 12 silver pennies can buy the same number of horses anywhere else in Middle Earth (most likely, they can't) and are thus even a fair compensation at all.
In summary, the moment you said "established currency system" you are bringing in a whole set of things you are not considering that makes the problem you think exists, not exist at all. Any sort of established currency system will be only on ONE standard - gold or government - and either of those can easily handle this problem. You just need to pick which one it is. 130