Your 15A breaker protects the wiring. All wiring downstream of that breaker must be capable of carrying 15A.
Your 6A switch can control a load of up to 6A. It is capable of opening and closing under a 6A load without damage, and carrying that load indefinitely.
Can the switch survive a surge for sufficiently long to take out a 15A breaker? The data sheet doesn't specify what fusing is required upstream. We can infer what surge the switch can handle from the data sheet from the incandescent/halogen lighting specification. These lights, when cold, have a resistance of about 10% of their running resistance, so draw a surge of 10x their running current. Although we note that this warmup surge tends not to take out a 15A breaker, otherwise the nuisance tripping would render them useless in lighting circuits.
In the absence of a specific fusing spec, you could be forgiven for assuming that it's OK 'with whatever is normally found in a house', like a 15A breaker. These switches will tend to use a TRIAC output, and these things are as tough as old boots. However, if your load is <6A, where's the harm in changing to a 6A breaker? You could also contact Lutron's technical department with a question.