Hi
I've used the hallmarks site to identify some badge makers before so it was just a thought.
You could try searching Graces Guide (no, not some online dating thingy....) and that might list firms that made buttons.
I don't collect buttons or badges per se but I do know that buttons were not an easy thing to produce if they were made traditionally. Clearly this firm did badges and buttons so of some note.
Doesn't Birmingham have a jewellery quarter museum? Records?
Hope you do find the answer!
