$stmt =~ s/TABLE_NAME.//g;    # doesn't work
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Just to note that the dot might cause major problems there, as it matches "any character".

If your table name is 'customers' and you have another table called customersummary, then you'll replace the first letter of that too, so if you're changing 'customers.' to 'buyers.', it will also change 'customersummary' to 'buyers.mmary'.