The Aussies had a 57 inch SMG sling in 1943 and could have slapped hooks on that to use with the Bren.
Having had a quick look at the British wartime training manuals I haven't found any references to the Bren being fired on the move, from the hip like, where a long sling would be a necessity. All the ones I've seen refer to it being fired from the bipod prone or from behind cover with it supported. The late war Indian army training manual with its bias towards jungle warfare does show the LMG being fired from the hip and the web sling for the Vickers-Berthier was long enough, more easily adjusted and had a much wider central section to spread the weight.
Are there any wartime jungle warfare training manuals that introduced firing from the hip for the Bren?
ATB
Tom