As Debra says, his record would seem to be among those that did not survive, it is certainly not to be found.
However, there is a medal record for him on line, his entry in the Register of Solders' Effects is to be found and his Commonwealth War Graves Commission entry is:
https://www.cwgc.org/find-war-dead/casualty/243380/brady,-/The Effects register tells us he died of wounds in No 21 Casualty Clearing Station (at the time in the town of Ytres (not Ypres)). The war diary of his battalion will tell you what they were doing at the time.
Do take the advice of scotmum if you contact the museum, it is unlikely that they have any more direct info for him.
MaxD