Despite the very generous comment from maypolebaz, I am about to prove that my knowledge is far from "encyclopaedic"!
As is generally known, a large number of Home Guards were transferred from 1942/3 onwards from normal infantry duties in order to man anti-aircraft batteries of various types: light and heavy guns and rocket launchers. By mid-1944 some 118,000 men had been transferred or conscripted, about 7 percent of the total service. In inland Britain these batteries were obviously concentrated in and around urban areas.
Anti-aircraft defences were the responsibility of the Royal Artillery. Many units had been formed by the outbreak of war. The transfer of Home Guards meant that many of those trained men could be transferred to more aggressive duties in the buildup to D Day and beyond. Some batteries were almost wholly manned by the Home Guard.
As with the Home Guard as a whole, records concerning these HG AA units is fragmentary in the extreme. Virtually all records were destroyed after the war. It is just possible, however, because it all remained basically a Regular Army function, that there is more information still tucked away somewhere in the National Archives or the Imperial War Museum. But nobody seems to have dug it out and, as would have been especially valuable, produced a list of what unit was located where.
So, in Birmingham, what seems to be known is based on fragments of information in people's memoirs. It does sound as though Brandwood House was the HQ of more than one local AA Battalion and the battery or batteries associated with any of these battalions were located somewhere in the neighbourhood. I have seen references to batteries on Billesley Common, in Swanshurst Park and also one described to me as "in Crabmill Lane in Kings Heath, near to the Maypole". (I should be interested to hear if the latter reference makes sense – I don't know that part of Birmingham and the only Crabmill Lane I can find seems rather further away). There may well have been more – located on any piece of open ground.
I have one story from the 71st Warwickshire AA Battalion with obvious connections to Brandwood House –
https://www.staffshomeguard.co.uk/DotherReminiscences74KingsHeath.htm - but, I'm sorry to say, absolutely nothing so far on the 69th.
Chris