Two-piece ball-buttons are very difficult to accurately time-date, unless there's a datable backmark. They've been manufactured with essentially the exact same form since approximately the 1830s. They are still being made today... see the VMI comment coming up momentarily. When they first appeared, they were used mostly on Civilian clothing, although some did get used on Military uniforms, and especially Military School uniforms. The cadet uniforms at VMI (Virginia Military Institue) still have these large ball-buttons. By the latter 1800s, they'd mostly fallen out of favor, except for schools uniforms, and some women's clothing. So, in view of all those facts, your large ball-button is most likely from the 1830s through 1870s or so.
The photo below shows a Virginia Military Institue cadet in full uniform, in 1910.