Assuming a technology level no later than 19th century, does the fact that it is not easily divisible and not necessarily fungible preclude it from being used as a currency?
No, it does not prohibit it, but it makes it extremely unlikely. For one, it is easily divisible; the same way gold was: A "bank" that has many rats can issue a promissory note in rat fractions that becomes money. They can also have something difficult to reproduce or fake on their promissory notes; much like our own money: Special paper, ink, and an extremely intricate design hard to reproduce particularly with late 19th century technology. Remember, our own early money IRL was a promissory note redeemable in silver or gold. Your is just redeemable in Rat! But it takes (for example) 100 single notes to actually get a Rat. For lesser denominations, they are still good for trade: The baker knows that he needs to sell 100 loaves to get a hundred singles, but at the Bank he really could trade those 100 singles for an actual Rat. If he wants. But he might prefer to let the Bank take any risk of accidental Rat death, and just keep the stack of promissory notes that are just as exchangeable. It is safer.
This is the flaw in your system, I think. It will be replaced by promissory notes almost immediately.
It is a misconception to think Scarcity is important for a currency. It is not exactly so. Money must be difficult to forge, and difficult to just find lying around in nature.
The scarcity of gold and silver just meant it was hard to fake, and hard to just go on a hike and get some. In real life, our currencies are hard to fake, but are not themselves made of anything scarce. And they don't just occur in nature, so it is hard to just hike in the woods and find some growing on a tree.
But that said, not everything scarce and hard to find makes a suitable currency: Consider Truffles. They can be extremely expensive, they can't be faked, they are hard to find, but they are an unlikely currency.
The defining characteristic of currency is widespread trust that it can be exchanged for equal value of goods, labor or services. Now "equal value" is nebulously defined, a fuzzy definition. Still we know what it means; that if a guy gets paid 100 singles for a day of labor, he knows what else he can buy for that 100 singles, in food, in entertainment, in tools or travel or transportation. He trusts that the five Twenties he was given are going to be accepted in trade for almost anything he wants to buy. The money does not have to be backed by anything as long as effectively everybody believes it will be accepted in trade; that is why vendors take it, and workers accept it as pay.
So the biggest concern about money is forgery and theft. If your Rats are very difficult to forge and not easy to find ; and no easier to steal than modern currency, then they can be a basis. But like Gold and Silver, IRL, they would likely be replaced by promissory notes, and then just replaced by hard-to-counterfeit paper products.
Added: This occurred to me later: You provide the perfect reason that Banks for rats would arise, how it is in their self-interest and why people would use them (in their self-interest): The rats sometimes reproduce. So the bank, in return for ensuring the life of your Rat (the promissory notes they give you, upon deposit, are redeemable for any Rat, not just your own), also keep any new Rats your rat may produce. Instead of our massive vaults for holding Gold and money, they have massive guarded rat gardens; with the rats in trays, perfectly sunned, watered, cooled or shaded, provided medical care as needed to optimize reproduction: That is their profit; it minimizes rat death and rewards them for being custodians of the rats. Like our own first banks, they can "loan" rats, by producing promissory notes collateralized by real property; e.g. a mortgage on your farmland so you can purchase animals, equipment and seed to farm it. Interest bearing loans, of course. People use the banks because they provide free (or low cost) life insurance on their rats, in return for the rights to offspring. Like life insurance, it is better to give up the rare lucky chance of offspring, to be protected against the unlucky chance of losing a rat in an accident, fire or assault.