Just trade him and resign him after the year.
That almost never actually works. If a player wants to stay, that typically means they don't want to be traded. Once a player is traded and the season ends, they're so close to free agency that they might as well see what spots are available and maybe earn more money. Typically, that's not with their old team. That's also assuming they don't have any hard feelings about being traded when they didn't want to be. Gotta remember, these are people, not robots, they have their own emotions and agendas.
From the team's point of view, if the Sabres want Greenway, it's better for them to just re-sign him. The pick that they could get for him is almost certainly not as valuable as the player himself, re-signed to a fair deal. Trading him opens them up to the risk of losing Greenway for a lesser asset (the pick) when they didn't have to, or re-signing Greenway for a higher dollar value this summer than they paid now.
It's a poor risk-reward proposition you suggest. There is a single scenario where they win in it (Greenway re-signs for the same amount
and they draft a useful player with the pick), but there are several scenarios where they break even but with unnecessary risk, or lose by various degrees.
Ah, almost forgot to say what I came in here to say. Good for Greenway! I always liked him, even if he did make me pull my hair out sometimes. I'm glad to hear his game has matured.