So I've been completing some Fitness shops for 360 relay. I'm not being awarded any points to any specialty badge for any of these shops and I've completed it seven of them. When I contact the I secret shop tech support they tell me that these are service specialty badges. However, when I contacted the MSP they tell me that these are classified as Fitness facilities not service. Is there a badge for Fitness facilities? If not then why am I not being awarded any points towards any specialty badge for completing shops through them?

You don't receive points towards a badge for all shops, just specific ones. It sounds like you are getting points towards the service specialty badge though. I also have done fitness shops that counted toward that badge. You will still receive at least 1% towards your next level though.