First published as a serialization in Cosmopolitan:

DALLA THE LION-CUB -- (Hearst's International Cosmopolitan Magazine, Feb-Apr 1924) was copyrighted 10 January 1924

Nash's and Pall Mall Magazine may have also serialized this story in July 1924. 

First published in book-form by Putnam in New York, likely in May or June 1924. 

I own a Putnam first edition with original DJ, and also own the Photoplay edition published by Grosset & Dunlap.

Hutchinson published its edition, under the title PERILOUS WOMEN, in London, with ads at the rear marked "Autumn, 1924." 

The Hutchinson edition contains two novels: PERILOUS WOMEN (titled THE GARDEN OF PERIL in the USA) followed by DALLA THE LION-CUB.