There is no specific adhesive to install asphalt shingles since they were designed to be mechanically fastened in place. I understand your issues about not wanting them to penetrate the inside of your sauna. Proper roofing practice would require nails. So, your options would be penetrating nails or installing another layer of ½ decking to give you a full 1" to avoid nail penetration. This is my professional recommendation.
Suppose you don't want to do it that way, and the shingles are just for looks. In that case, you can reach out to a company like
Henry, who makes adhesives, and see if their 900 construction adhesive is used in roofing may work. I don't think it will work on the barrel. You will not get any guarantees it will work, but it may be worth a try for what you want to do. This is their contact number: 1-800-486-1278 and their email:
productsupport@henry.com.