Unless he plays in the finals, KT in my opinion doesn't belong on the Cup. He shouldn't get it for "long and meritorious service" in the NHL for other teams.
However, I think the Hawks played a marginal first game and won, and they will play significantly better in game two and it will be difficult for TB to win. If the Hawks go up 2-0, they will likely win the Cup in four, maybe five games. In that scenario, KT probably gets played in one of the final games and then his name goes on the Cup.
Regarding who gets passed the Cup, likely he gets it first rightly or wrongly. A lot of key Hawks seem to get the Cup passed to them quite late, and the first that comes to mind is Kane, who always seems to be an afterthought in receiving the Cup.