Most definitely depicting riding to hounds. I've been fortunate enough to do that a couple of times. The dress code is very formal, and buttons could easily be lost. That's a nice find. The hounds look strangely like poodles though.
I have went on a couple fox hunts . Some carried whips. One carried a Trumpet type horn. Not for me i like my western type saddles. Some of those type people are amazing horse people . Still is a amazing button I could find no motre info on it sorry