If I'm not mistaken, players can appeal but a team cannot?
What I am thinking is this: Brady will appeal, and ask for arbitration rather than Goodell handling the appeal. At that point the whole "independent investigation" and subsequent mockery of a report will be exposed as a sham. That will result in Brady's suspension being lifted.
Once that is done, the Patriots should appeal their punishment, because the Wells report was a primary part of the reason for the fine and loss of draft picks.
The only thing Goodell would have left for rationale would be the Patriots being "uncooperative". The Pats could counter that by saying that the findings of the (truly) independent arbitrator support them - hopefully to a point of exposing the whole process as a sting and a witch hunt. However, I think Goodell would still stick with that alleged lack of cooperation to justify the egregious penalty - but it would be worth a shot.