Matt,
You are probably finding that online information about the Home Guard is spread pretty thin when you're trying to pin down particular units, locations and individuals. Sheer luck now plays as much a role as any in digging out information like this.
A few general points about the Home Guard which may help you to home in on your grandfather.
Volunteers were invited to come forward from 14th May 1940 and hundreds of thousands across the country did so almost immediately. Many had seen Great War service. There were ten Battalions initially in Birmingham but within 7 or 8 months these expanded to over 30, each containing anything between 1000 and 2000 men. All of these Battalions operated in an infantry role and the men trained in all the disciplines necessary to produce a good infantryman. A man volunteering for Home Guard service would normally serve either in a unit adjacent to his home address or in a works unit at his place of work.
In January 1942 men started to be directed into Home Guard service and joined the vast throng of volunteers. During 1942 the Home Guard took over many anti--aircraft duties, thereby releasing Royal Artillery men for other work. Most of these gunners transferred, reluctantly, from normal Home Guard infantry units to a battery in their neighbourhood, located in a park, on waste ground or at a similar site. Others entered this a-a service direct as a result of their conscription. None would of course have had previous experience and required training to fulfil this new role. The a-a units were totally separate from the conventional Home Guard units and bore different titles. They might have comprised batteries of conventional heavy guns or the newer, z-rocket missile batteries
There is a reasonable amount of information available about individual, conventional Home Guard units but not a lot, unfortunately, about the a-a batteries. There is information about the various Birmingham units in my website (link below) but, somehow or other, you will need to make some guesses as to when and where your grandfather started his Home Guard service. Within the website there are also some quite good descriptions of the operation of z-batteries although, unfortunately, not sited in Birmingham (use the website search box and a subject of "z-battery" to find them). And of course, we're not yet quite sure whether your grandfather served on that equipment or on more conventional guns.
Welcome to the Forum, by the way.
Chris