All that can presently (November 2023) be gleaned about Harry Bulger Jr. from sheet music covers and by Internet searching is that he was presumably an American actor/singer who has been featured on just one work namely Wait and See (1919) music by Charles R. McCarron, co-author Carey Morgan, words by C. Helene Baker and published by Jos. W. Stern & Co., New York, USA
His portrait on the left is taken from that same work.
Harry Bulger is presumably his father.
Surfers are welcome to contribute any further information about this performer.