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How would they sanitize the clothes they already gave them when they leave and return?

On the way in, they leave everything from outside, and on the way out, they leave everything from the inside and take what they brought in from the outside.

Because it's the military so everything technical is done/designed obtusely...although your idea is more in line with typical financial/resource wastefulness so it wouldn't surprise me either!

How do you ensure the new clothes aren't embedded with some chip placed by a spy who has access to the supply chain but not the black-site?

I'd assume they give them new clothes & microwave said new clothes. along with other precautions.