Cats and rats - - sorry - - no luck there. Cats belong to a different labor union from rats. I have two cats and when a rat joined the crew, the cats where not even mildy interested. I used traps, have-a-hearts, glue paper, and poisons and cats. Result - fat rat and lazy cats.
One early morning I awoke with one of the cats curled up by my tummy. Opening an eye I spied the rat lazily preening himself on a shelf across the room. I poked the cat in the head to wake him up. I pointed at the rat and said, him rat, you cat, get him. No such luck the cat yawned and went back to sleep.
In reality, the rat lives and does his damage in the bilges where the cats cannot get to. So even if they wanted to go after the rat the rat can duck down crevises and gaps that no cat can navigate. So use the various traps, glue papers and poisons.
Side note to prevent the rat from gaining entrance in the first place is best. I purchased some of those vinyl covered wire shelving from home improvement/hardware stores and cut out shapes to make "grills" over all the vent holes, and any other opening from the inside of the boat to the outside. The shelving had a wire mesh pattern of 1/2 " by 1" and was perfect. It is amazing how many opening there are in our vessels for engine ventilation, etc. All of these are where the little monsters can gain initial entry.