print "Name: $dbh->{Name}\n"; returns Name: prod_01 which is the database name.

I'm hoping I might be able to determine the hostname used for the connection (which one of the nine hosts listed in the tnsnames.ora definition was used for this connection).

That's what I get for reading too fast. How about trying a get_info call on SQL_SERVER_NAME? I've verified the following against an Oracle database I have access to, but not in a multiple hosts/load balance context.

#!/usr/bin/perl
use strict;
use warnings;
use DBI;
use DBI::Const::GetInfoType;

my $_db_name = 'host';
my $_db_user = 'name';
my $_db_pass = 'password';

my @_dbi_path = ("dbi:Oracle:$_db_name",$_db_user,$_db_pass);

my $handle = DBI->connect(  @_dbi_path,
                            {
                                PrintError => 1,
                                RaiseError => 0,
                                AutoCommit => 0
                            }
                        );

#print join "\n", sort keys %GetInfoType;
print $handle->get_info($GetInfoType{SQL_SERVER_NAME});
[download]

See DBI::Const::GetInfoType and get_info in DBI.

Update: Still no dice -> in DBD::Oracle, this is just a synonym for $handle->{Name}, as is SQL_DATA_SOURCE_NAME.