You could sign up for the free trial of Prime, claim the game and cancel the trial. Unlike PS+ or Gamepass, the game is yours to keep even without the membership. Or so I understand anyway.
I feel you. Think of it this way: by signing up for a trial and claiming a license for these games, you are costing Amazon money. That's the only reason I have an Epic store account.
Once you claim this game, the game is yours to keep forever. If you cancel your Prime membership, you will still be able to download and play the games you have previously claimed in the Amazon Games app. If you would like to claim new games, you will need to have an active Prime membership.
As PlasticExistance said it's free if you jump through some hoops. Also I tag them all with Prime so if you use a Lemmy client which can block strings you can ensure you never see any of the Amazon giveaways.