The earliest date the OED has for “Anglo-Manx” is 1934. They do not even pick up on the title The Vocabularly of the Anglo-Manx Dialect published in 1924.
In 1895, Brown felt that “I have an idea that Mr. M[oore]’s new book will show plainly that we have arrived at the last squeak of the Manx language proper.” He continued in his letter to Egbert Rydings who had written short stories in Anglo-Manx:
“So I think what we have now to do is to make a new start, making Anglo-Manx dialect the basis. In its turn this will probably become obsolete, but meanwhile the catastrophe will be deferred by your stories, and, perhaps I may add, mine.”
However, I can trace it now to John Rhys writing in 1892 (not that much further in this case) in “Manx Folk-Lore and Superstitions. ii.” Folklore iii.1 (1892): 74–88. On pages 76, fn. 1 and 85 to be precise and it is used in a linguistic and not an ethnic sense.