Yes, querying sqlite_master is another way. I used this formulation last time I did this, probably an even better formulation is possible:
my $check_if_bracket_table = $dbh->prepare("SELECT name
FROM sqlite_master
WHERE
type = 'table' AND
name = 'Brackets'");
$check_if_bracket_table->execute();
my $ref = $check_if_bracket_table -> fetchall_arrayref;
if (@$ref >0){...} # the Bracket Table exists...
Update: the sqlite_master table contains a lot of meta information about the DB. A lot more can be discovered than just whether a table exists or not. I am not an SQL guru, this is something I did last year to solve a problem. I had to do some stuff with the Brackets table, if it existed. Otherwise there were some rules about how to deal with situation.