Need new roof for soundproofing

I am desperate to find someone to install a roof over my shed style roof on a vinyl sunroom by Champion. I have solid foam core roof panels and when it rains, it is so loud. There is nothing wrong with it now but I dont want to create a problem having the wrong roof put on. I dont want to nail in a shingle roof. Any suggestions?
My thought (probably delusional) would be to adhere some wood strips with glue and mount plywood as a base and then shingle and flash the heck out of it where it meets the roof. The pitch is fairly low so not sure if this is the best option. HELP

Remove the sunroom and replace it with something that has a real roof.

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Adhere polyisocyanurate insulation and adhere rubber over it would be the only real option.