The thing about genes is that if there aren't natural selection pressures to keep certain traits, they tend to drift, and traits can change or become entirely non-functional over a surprisingly short time. I would expect, unless there was a concerted effort to maintain the radiation detection trait over those 10,000 years through careful breeding, the cats would lose their radiation triggered appearance change behavior before it actually had a chance to be useful to people.
Maybe genetic engineering could advance enough that they could make it so changes to the gene would lead to sterility or something like that. Like make it part of some essential reproductive genes.
Admittedly I'm thinking about this more in terms of sci-fi worldbuilding than realism.
In 2004, a Danish company created mine-detecting plants that changed color when their roots were exposed to landmine leakage. So yeah, this actually makes sense that we'd imagine this kind of solution.