->execute($value) does not return the rowid.

See last_insert_id of DBI on how to use ->last_insert_id. Also, you can use the RETURNING SQL clause:

my $sql = <<"SQL";
    INSERT INTO $table ($column) VALUES (?)
    RETURNING id
SQL

my $sth = $dbh->prepare($sql);
$sth->execute(...);
my $rows = $sth->fetchall_arrayref( {} );
say "New id: " . $rows->[0]->{id};
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