Salesman #1: You're crazy with the heat. Credit is no good for a notion salesman.
Salesman #2: Why not? What's the matter with credit?
Salesman #1: It's old fashioned. Charlie, you're an anvil salesman -- your firm give credit?
Charlie: No, sir!
Salesman #1: Nor anybody else!
The Music Man, "Rock Island"
Writing is believed to have been invented originally for goods logging and tracking for commerce. I believe your best bet would be for your society to have bypassed directly over "hard currency" to credit.
The problem with hard currency is the need for it to be intrinsically valuable. A bauble of silver for a cubic meter of seaweed, or whatever. The problem, as you state, is the unlikelihood of metal working to be feasible underwater.
This is a different dimension of a problem experienced by the Venicians in the Middle Ages; responsible for handling thousands of "pounds of silver and gold" worth of trade, they needed a method to keep track of goods without actually counting out hard money over and over again.
They invented the cheque. (Muslim traders used a form of cheque as well, and JewishHistory.org claims Jewish traders invented the cheque. The Dutch Republic in the 1500s had professional "cashiers" who handled cash for citizens, charging a fee; they provided additional services, including cash-on-demand to anyone bearing a written order. But I'm going to credit Venice today.)
Cheques, promissory notes, bills of exchange; these negotiable financial instruments were all about notating a large value worth of goods onto some recording medium, and then using that recorded value in place of the goods themselves. The bearer of the instrument could then extract the value on demand.
This was the bases of money in the world from the middle ages 'till Nixon killed the Bretton Woods system, and Federal Reserve Notes could no longer be converted to hard currency. Yes, pre-Nixon you were still spending gold and silver, you were just using paper certificates to do so.
Your underwater civilization just avoided the hard currency middle-man.
From the beginning, they either did hard-barter with goods, or traded negotiable financial instruments.
Perhaps they started using shiny rocks, or pearls, or volcanic rock pulled up by "miners" who could swim down to the lowest depths of their ocean, but for underwater critters these goods would be even less convenient than they would have been for us.
So from the beginning, they would have come up with a system of trade involving negotiable instruments, payable to the bearer on demand. A group of critters farming fruit would trade an invoice for fruit in exchange for some good they needed; that instrument would be further traded until somebody needed some fruit, at which point they'd bring the instrument to the fruit farmers to demand their goods.
Now, negotiable instruments are much easier to counterfeit, in theory, than hard currency, which is one of the many reasons why hard currency remained popular until even today. So your critters would need some way to validate these negotiable instruments.
- Do they have a form of writing? This would essentially emulate the early banking systems on our world; an instrument would be written by the possessor of a good, and countersigned by trustworthy individuals to prove the value of the note, which could then be traded freely.
- Even without writing, verification is possible. The "split tally" system was very common in the middle ages amongst predominantly illiterate society. A stick would be scored with a system of notches showing taxes owed and then split lengthwise; each side of the deal would receive one piece of the stick. You could then validate the document by putting the sticks together; while you could modify your stick, you couldn't modify the other stick without possessing both, proving that some change had been made.
Here's a really, really crazy idea: Underwater blockchain. Bitcoin works by keeping a long, immutable ledger, distributed around the world known as the blockchain. The mathematics involved allow you to prove the correctness of a ledger. New transactions are added to the blockchain in such a way that they become a permanent records. What if your critters used something similar to record transactions? ...and what if the blockchain was actually whalesong? Let's assume that whales on your planet are not sentient... or at least one species known for it's song isn't. A group of your critters learn how to interpret the whalesong, and how to encourage these whales to modify the song. Let's handwave that the whales have "photographic" accoustical memory, that they refuse to change songs they have learned, but learn new songs easily. You now have an underwater banking system whereby transactions are recorded by "tellers" who literally "tell" the transaction to a local whale, who then weaves that transaction into the songchain it sings. Other whales hear the new song and share it. Anyone who wants to retrieve a transaction "hums a few bars" to the nearest whale, who then gladly sings the rest of the song. Because songs can only be appended to, never modified (the handwave), by listening to the song you can verify that the bearer of a financial instrument actually bears the legitimate instrument, and can then provide them with the good (logging it into the songchain, of course).
We can further stipluate that whalesong is generally at the lower range (or upper range) of the frequencies our critters can hear; in general, it's just the normal background noise of the ocean. But a specially trained critter can listen to the chainsong and interpret it, and encourage local whales to append to the chainsong. The whales just do what they have always done: share songs with each other. Their song carries for thousands of miles in the ocean, ensuring that every transaction is eventually recorded everywhere in the world, allowing you to "spend" your instruments anywhere where the song has been recorded.
Now, I'm sure somebody with a background in information theory could prove that the type of whalesong which travels such long distances conveys too little information to be able to log EVERY transaction on an entire planet, but if we're willing to suspend disbelief a little bit, the songchain would only be used for large, interocean transactions. Local networks of trust would be good enough for day to day operations, with enough signatures on an instrument or a tally-shell business works fine. Long distance trade could be handled by songchain however.