Your button has the logo used by ‘Mark Cross’, founded in Boston in 1845 by Henry W. Cross and named after his only son who later inherited the business as ‘Mark Cross & Co.’. Originally they were a harness, saddlery and trunk-making company, producing “the finest leather goods for the horse and buggy rider”.
One of their employees, a salesman called Patrick Murphy, persuaded Cross to expand the business to include small luxury items such as purses, wallets, and notebooks. It was a huge success, Murphy got promoted, and he ultimately bought the business after the death of Mark Cross. In 1892 Murphy moved the business to New York and opened a fashionable store on Broadway. By 1893, he had two stores in New York, had also opened a store back in Boston and another in London, England. The business rapidly expanded to become the quintessential American luxury lifestyle brand, with more than 20 stores selling imported luxury goods including fine china and crystal as well as leather items.
They also sold clothing including: dresses, jackets, skirts, trousers, and sweaters for women; suits, jackets, trousers, knitwear, and shirts for men. I would assume the button relates to one of those items. With a ‘Birmingham’ backmark, I would guess it's from something made (post-1893) in England for sale in the London store or for export to the US for sale there.
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