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Partially solved: I have access to CA's root certificate and I am able to validate the certificate in JWS (openssl verify) as well set set is as "client certificate" on the server for mTLS. Still, the correct approach for JWS remains a mystery to me.
On first thought, that is a sensible answer. I was not aware this could be the case so I have omitted the fact that the another service is a physical machine where certificates cannot be updated remotely by design and nothing can be updated on their end (they, as an organization, can use their root cert though). Also, I don't really understand how in this scenario they could trust me but not any other web service that has a certificate signed with Let's Encrypt. I understand they want to remain in control whose certificates they sign. Should I update the question or this is irrelevant?