As I've mentioned in other discussions of buttons, the style (or "font") of the lettering on a letter-button is often very important in the quest to correctly identify it.
Although this button's backmark is a railway-supply company, which would "suggest" that the CP button is a railroad button ...I note that the button's "CP" lettering is Old English style lettering. Very few railroad buttons have Old English letters. But antique-eraPolice buttons commonly have Old English lettering.
Furthermore... although the backmark is American Railway Supply Company, that company did not actually manufacture any buttons. The railway-supply company purchased its buttons from a major button-manufacturer -- which will produce buttons with whatever front-design and backmark its customers desire.
Sometimes, when a button-customer goes out of business, unused button-backs with that company's name are "left over," unused, at the button-manufacturer. Rather than throw them away, the button-manufacturer will use them to fill other orders. Button-scholars know that has happened many times. A well-known example is US Army eagle-buttons which have a South American company as the backmark. (Those eagle-buttons were manufactured by Scovill, which apparaently used some leftover button-backs to fulfill an order from the US Army.)
In summary... my guess is that the CP button, with Old English lettering which is very uncommon on railroad buttons, is quite probably a City Police button, made with leftover ARS Co. backs after that company went out of business.