LOL. Ironically, wearing the vest on top is what we would call "cowboy."
As to the only other thing I can think of that would be close to dressing as a cowboy might be the hats. The closest ones I know of in a police uniform would be the ones that most State Highway patrols and the RCMP wear.
There you go - we are very proud of our police. They are brave and distinctly professional - I would certainly prefer them to the paramilitary types we see in the US.