It was given to people who only had an early stage of the disease, as per Mikkel in Amalienborg, so it shouldn't have been apparent yet whether they were simply dying or trollifying. At least during the creation of the cure, the only indicator seems to have been the resilience of the infected.
I keep thinking of the safety measures in Mora - they had cats sniffing the luggage there, and it feels more likely to me that they were checking for traces of infection, e.g. in an infected person's clothes, than for, say, Rash beasts that got into the luggage or that someone might have tried smuggling into the safe zones. So I'd guess they can detect the Rash itself, not merely trollified creatures.
In the absence of Word of God, I'm going to offer a different point of view.
I'm assuming that by "the Rash itself" you mean infectious particles outside a host body -- though please correct me if I'm wrong.
Minna has said the Rash agent (virus, bacteria, prion, fungus spore, or ???) doesn't survive long outside a host. That's why Lalli, after his first scouting mission, had his clothes put in the Cattank's ultraviolet-light disinfector -- though he could have just been made to stand in the sunshine for a couple of hours. The Mora government washes and sterilizes the clothes people are
wearing -- though we didn't see how they handle
shoes and accessories such as jewelry, hats, hairclips, wallets, etc. But it doesn't extend that treatment to the contents of bags, which are merely inspected. (It seems like either the Mora authorities or Minna herself didn't consider the possibility that someone might have taken off something s/he was wearing and put it in a bag -- or else they assume the infectious agent never outlives the check-in process.)
So to my mind, the small amount of Rash contaminant a person could pick up in casual contact with a infected environment is not what the cats are there to detect.
HOWEVER, all non-feline mammals can carry the Rash if they're not immune. In some ways, the smallest mammals (mice, voles, etc.) pose the biggest infection risk because they can creep in anywhere. I think the cats are there to detect the threat of live but infected critters sneaking into baggage (and/or very small Rash-beasts if you don't consider those literally "alive") -- as well as people/pets/livestock who are infected (the way Kisu reacted to Tuuri).
This still leaves the possibility that a malicious or crazy person might try to smuggle in a chunk of a Rash-beast. (Is anyone in the Known World naive enough to think of a dead, preserved Rash-creature as merely a trophy, like the "shrunken heads" soldiers brought home from the South Pacific?) We don't know how long severed tissue stays infectious; it does at least seem to lose the ability to move and/or regenerate once the Beast/Troll is killed. It seems plausible to me that cats could detect that, too.
