DBI provides a last_insert_id method as of v1.38, but I haven't had luck in getting it to work, at least with my DBD::mysql version (2.9003).

I applaud your goal of DBD-agnosticism. But it's really hard to be DBD-agnostic without also being DBD-specific. Look at all the big cross-platform DB modules out there (Class::DBI for instance). What they do is have all DBD-specific features subclassed out. That's the "right" way to do it. In my own database module, I had to resort to something similar for getting the last-insert-id, as well as schema detection. It goes something like this:

package Foo;

sub new {
    my $class    = shift;
    my $dbh      = ...
    my $subclass = "$class::$dbh->{Driver}{Name}";

    eval "use $subclass; 1;"
        or croak "$dbh->{Driver}{Name} is unsupported";
    
    bless { dbh => $dbh }, $subclass;
}

sub dbh {
    $_[0]->{dbh};
}
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Then you have a few small DBD-specific modules implementing insert_id:

package Foo::mysql;

sub insert_id {
    my $self = shift;
    $self->dbh->{mysql_insertid};
}

###########
package Foo::Pg;

## notice how Pg requires the table and column, while the
## other DBDs ignore these args

sub insert_id {
    my ($self, $table, $col) = @_;
    my $id;

    eval {
        my $seq = $table . '_' . $col . '_seq';
        my $sth = $self->dbh->prepare( "select currval('$seq')" );
        $sth->execute;
        ($id) = $sth->fetchrow_array;
        $sth->finish;
        1;
    } or die;
    
    return $id;
}

##########
package Foo::SQLite;

sub insert_id {
    my $self = shift;
    $self->dbh->func('last_insert_rowid');
}
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(In my own module, I don't actually bless the objects into the subclass, but I do store the name of the DBD subclass and call insert_id as a class method when I need it) Subclassing is clean and easy to maintain, plus it encourages me to stretch support to as many DBDs as possible.