Crypt::RSA says the key argument is a "Private key of the sender, a Crypt::RSA::Key::Private(3) or compatible object." On the other hand, you're passing an unblessed scalar, which has no methods, rather than an object with the API Crypt::RSA expects.

You'll probably want to use something like
$private = [doc://Crypt::RSA::Key::Private]->new(filename=>$private_key_file)
to load the private key, and then pass that object along to sign().