Jill is looking for the narratee in a game - why can't you continue using the word, Jill? While I agree that games are not narratives, I will claim that they are narrative environments with the potential of creating stories, as are our real lives. Or do you, by game-narratee not mean the person to whom the story is told, but the player the game is aiming at? In that case, I feel that I come a fair distance through distinguishing between the player and the character, and then use not the narratee but the reader-concepts of Eco to suplement the implied reader of Iser - only I change reader to player: model player, implied player, target player, actual player...
Looking at Chatman's model again: no, I think you need to distinguish between the two position of the player in the game: the In Character position being that closer to the narratee: the character or the avatar, the Out Of Character position being the one behind the game, the player.
This might not be the graceful word you are looking for, Jill, but I like the word "player" because it implies a person active in relation to the game, one capable of playing not only with the game but with the context of the game, and able to be a subversive player as well as a submissive player - all the reader- strategies and then some. This is also good because the reader, while being given the position of power inherent in being the one who realises the work into text (Barthes), does not hold the active position of the player of games, while the player is allowed to create, to make up, to add and subtract elements within the rules more on the line of a performer or an author. And with the performance aspect of "play" I feel that "player" is even more relevant.
In both reading and playing, the narratee is the one being manipulated, being told (and the reason Implied Player doesn't fit is that the implied reader and the narratee isn't the same, you know that), and so the character or the avatar are good distinctions from the player.
The more I look at this, the more I feel that player is a good and elegant word in this context. While I'd love to make up a word and have the world use it - there's a good heap of words out there which still work.