Can I use PGP to sign a message without providing cryptographic non-repudation?
Yes, there's an indirect way to perform what's asked with PGP and variants.

Draw a an asymmetric public/private key (of a type that can sign), protected by a passphrase. Publish the passphrase-protected private key. Discard the public key (it's in the private key anyway). Use the passphrase as you would use a symmetric key: share it by trusted means between sender and receiver (perhaps together with the passphrase-protected private key). Generate, transmit and verify a detached signature (sig file) of a file to integrity-protect as you would do for a MAC of the file.