$DH->priv_key($DH_private);

When you call priv_key() like that, I believe you provide the function with two args - the string "Crypt::DH::GMP" and $DH_private.

However, the priv_key() function takes only one argument, and returns a string.
I presume you therefore need to be calling priv_key() as something like:

my $string = priv_key($DH_private);
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Afterthought: Or perhaps you'll need to call it by its fully qualified name:

my $string = Crypt::DH::GMP::priv_key($DH_private);
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However, I know nothing of this module's functionality, and I couldn't find any example of priv_key() usage in either the documentation or the test suite.
AIUI, the priv_key function is provided in xs/dh_gmp.c as:

char *
PerlCryptDHGMP_priv_key( PerlCryptDHGMP *dh )
{
    return PerlCryptDHGMP_mpz2sv_str(PerlCryptDHGMP_PRIVKEY_PTR(dh), 1
+0, NULL);
}
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Update: And all that essentially does is express the value contained in PerlCryptDHGMP *dh as a base 10 string, and then return that string (packed with extra 0's as needed).

Cheers,
Rob