When I try running my script with the following test set , it does not print anything in the output files, instead prints the data onto the screen!

This is most likely because with

    my ($name) = /^([^\n]+)/;
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you're extracting the entire header line as $name to be used as the hash key, but you have not set up your hash accordingly with keys such as "001.b1 gnl|ti|10009".  In this case, none of the select FILE1 statements would execute, so STDOUT (connected to the screen) remains the default output handle for print.

Note that ^([^\n]+) would match everything from the beginning of the record up to the first newline, because the character class [^\n] says "match any character but a newline" (the ^ within the class negates).

Try one of the other suggestions, or adjust the regex as needed.