This isn't really an answer but maybe a start. I spent some time debugging it. I couldn't replicate your error; however, I did have a problem with getting $sth->dump_results(); to work---it wouldn't work for me. Another other problem was that I had to set the env var for my dbi_driver. Maybe this will help.

#!/usr/bin/perl 

use strict;
use warnings;
use DBI;
use DBD::TSM;

$ENV{'DBI_DRIVER'} = "DBM";
$ENV{'DSM_DIR'}    = "/opt/tivoli/tsm/client/ba/bin/";
$ENV{'DSMADMC'}    = "dsmadmc";
$ENV{'DSM_CONFIG'} = $ENV{'DSM_DIR'} . "dsm.sys";

my @drivers = DBI->available_drivers;
print join( ",", @drivers ), "\n";

eval {
    my ( $server_name, $user, $password ) =
      ( "enigsy-ser-backup-01", "user", "password" );
};

my $dbh = DBI->connect(
    "DBD:TSM:enigsy-ser-backup-01",
    'user',
    'password',
    {
        RaiseError => 0,
        PrintError => 0
    }
) or die $DBI::errstr;

my $sth = $dbh->prepare("INSERT INTO foo VALUES (1, 2, 3)");
$sth->execute() or die $sth->errstr();
print @{ $sth->{NAME} }, "\n";
$dbh->disconnect or warn $dbh->errstr;
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