I tried GAPS first and a few months later I tried paleo autoimmune protocol. I have Hashimoto's and don't have major digestive issues. I found PAP to be way too restrictive and I wasn't seeing any noticeable changes in my Hashimoto's. I decided the only way I could do such a restrictive diet would be under the guidance of an experienced health care practitioner, someone like Chris Kresser. Until I find someone I trust and can afford I'm just going to stick with primal.
Some people try the PAP and feel better almost immediately, I think if that were the case with me I would have been willing to continue, as it was I gave it 3 weeks. There are a lot of similarities between the diets you listed so I would just pick one that you think you could stick to and give it a try for 1 month.
Life is death. We all take turns. It's sacred to eat during our turn and be eaten when our turn is over. RichMahogany.