No progress yet on this subject. I decided to forego the Mitchell and contact GCC and ask them if they knew anything. Got a reply back - contact the archives - that's right, the Mitchell. Going round in circles now.
One suggestion by the staff at the Mitchell was to try to find out who the factor is for that tenement, and they might know something about why the Coat of Arms is there. Apart from knocking on doors, I thought I'd ask on here first if anyone has any idea who they might be?