Welcome to the forum WillM.
You have picked an interesting area of collecting. I have a soft spot for martingale badges, bit boss badges and anything military horse furniture related to be honest. Initially I was inspired by a wonderful article in the Soldiers of the Queen publication from the 1980s (I have it somewhere about me but cannot lay my hand on it immediately) but it documented the various bit boss badges worn by the Household Cavalry and the Cavalry of the Line. Some of the ornate ones only being worn by officers, most other ranks being entitled only to a generic pattern. It is not a hard stretch from these to martingale badges!
Cheers
Mike
The Quartermaster)
www.qmsmilitaria.com