Yes, it was not easy but I've finally found the letter. :-) I've found it in the German book "Alexej, der Sohn des letzten Zaren" (Alexei, the son of the last Tsar) by Elisabeth Heresch.
Alexandra sent the letter to Alexei on October 31, 1914. As I only have the letter in German, here's my translation (only the part in which Alexandra mentions Joy):
"...What about your dog? He follows (your command) better when you call him "Joy". That means "joy" (the original word here was in Russian) in English."
Here is my copy in German:
"...Und dein Hund? Er folgt besser, wenn du ihn "Joy" rufst, das heißt auf Englisch "Freude"."