This is a matter of usage and not of grammar. "One" sick cat does not mean the same as "a" sick cat. "One" sick cat is used to mean that the cat is very sick. Generally, we don't say the number when speaking of one item unless we want to emphasize the quantity. In this case, It's not even an emphasis on quantity but, rather an idiomatic phrase.
Unless you have several cats and one of them is sick... Then you might say "I have one sick cat". Also, say someone says "I think your pesky cat scratched my chihuahua in a fight" and you answer "I have one sick cat - no others that could injure your dog". It may not be common, but it's not, in other words, impossible to think of a time when you could say "I have one sick cat" in English. So it should be accepted.