Alberta, I've got a feeling that the RO you spotted was Birmingham North. Don't know how long it was there for though as, like your Grandmother, Dad's death was registered in Birmingham in 1964. I wonder where mine was registered as I was born in the nursing home at the top of Hunton Hill. Will check.
I had a look at RO's in Birmingham and on Freebmd the following were listed: Aston (to 1924) Birmingham (to 1924) Birmingham North (from Dec 1924 to Sept 1932), Birmingham South (from Dec 1924 to Sept 1932) and Birmingham (from 1932).
Don't need to check now as I was born after 1932.