I personally know of no direct evidence that answers this question, though others might. Some additional ideas, which may have occurred to you already ...
Rosanna Walker's "Birth of a Lost Boy: Traces of J.M. Barrie's Peter Pan in Willa Cather's The Professor's House" provides some context on Cather's career in relation to the play's opening in New York, November 6, 1905. No evidence of Cather seeing the production is provided, although Mark Twain's response to seeing it is quoted there.
Peter Pan on Stage and Screen, 1904-2010, byBruce K. Hanson might be worth a look.
Second Star to the Right: Peter Pan in the Popular Imagination (see page 117) lists the cities where the American tour of Peter Pan went in 1906, and Pittsburgh is included in that list. (Cather moved from Pittsburgh to New York in the Spring of that year.) This section of Second Star to the Right is in dialogue with Fifty Years of Peter Pan, by Roger Lancelyn Green, which might prove useful.