I would say no. From what I've read his contract is back loaded to the point that no one would trade for him unless he agrees to restructure. If he refuses to restructure his deal then he will most likely be released and will get to chose his next team instead of having it thrust upon him. There are only 3 reasons I can see him agreeing to restructure and all 3 are fairly remote possibilities:
1. If he doesn't, the Cardinals won't release him, just keep him as the backup.
2. He really wants to play for 1 particular team, and that team just happens to be the one that trades for him.
3. To help the Cardinals. Hard to justify being a team player when the team is getting rid of you.