There has always been discussion about the "Secret" Grand Duke Ernst visit, and indeed, it is a major part of both the play and the movie "Anastasia," but in my research, the closest thing that I have ever gotten to confirmation of it are the letters which Ernst sent to Alexandra to negotiate a peace, which she rejected. A copy of ALexandra's letter stating "no peace is possible" is still in the Grand Ducal Archives of Hesse. She then sent the letter on to her sister Elizabeth, who took it upon herself to become involved.
After this rejection, Emperor Wilhelm tried to reach Alexandra through Ernst himself, by getting Priness Marie Vassiltchikova to smuggle a letter in from Ernst via Grand Duchess Elizabeth (who had herself used Marie Vassiltchikova in 1905 to get a letter to her brother Ernst)
Nicholas was infuriated by this meddling on Elizabeth's part, which could have placed the Empress in the position of being accused of treason, and he sent Princess V. into internal exile on her sister's estates at Chernigov. This information may be found in GARF, "Materialy k Zhitiu prepodobnomuchenitsi velokoi knyagini Yeizavety; Pis'ma, dnevniki, vospominaniya, dokumenty." and also in Bokhanov's "Sumerki Monarkhii"published in Moscow in 1993. You can find a good telling of the story in Hugo Major's Elizabeth, Grand DUchess of Russia.