"Джой" as quoted in a "Letter from Tsarevich Alexei Nikolaevich to his Mother, Empress Alexandra Feodovrana, 9 September 1916, page 323, Item 538, in the section "Family Life," of "Nicholas & Alexandra, The Last Imperial Family of Tsarist Russia," Harry N. Abrams, Inc., Publishers, 1998. The original letter is in English, with the Heir giving the cyrillic characters for the English pronunciation of the name of the dog. This appears to be the copy of the letter you illustrate.
IMO, you are safe to use the English "Joy," if you are intending to rename the dog you mention. Regards, AP.