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...
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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.