So, in my post apocalyptic world, trade and commerce is very important to many towns economies, and so is currency. They need to be able to trade with nearby settlements, but their are robbers and raiders, so you can’t travel around with gold unless you want to get killed. So, in the post apocalyptic North America, merchants have come up with a new solution. So, people have opened up these businesses were they keep your gold for a small fee, and the businesses give the people slips of thin tree bark with a signature and amount of gold on it, that they can use to get their gold back. My question is, could this system work?
Do you realize that this is how banknotes started? In effect a promissory note.
The value of the piece of bark would depend on both the credibility/counterfeit-proofness of the physical token and the reputation of the issuing business. If the business is distant, there will be a discount on the face value.
And perhaps reputable businesses/banks will start to recognize each other's notes, and they get even more practical. One could swap a large note for several smaller ones at the local bank to settle a debt.
The idea that only central banks may issue notes is a more recent development.
Paper money and promissory notes were accepted in the Middle Ages because the guild of moneylenders seriously monitored their members. If one of them couldn't cover the loses, he was executed summarily (by beheading in the Kingdom of Aragon of the 14th century).
I think it could work the same in a your scenario: if there's someone watching over those merchant-banks so they don't run away with the money or lend more than they have, people will trust them with their savings.
So, how independant are your merchants?