If you don't care about the child's STDERR, you can send it to the parent's STDERR (using '>STDERR' for open3's third arg). Then, all you need is

...

while (<$stdout_fh>) {
   chomp;
   if (/Have a nice day!\z/) {
      print("$query\n");
   } elsif ($answers{$query}) {
      print($stdin_fh "$answers{$query}\n");
   } else {
       # ...[ Do something with unrecognised output from STDIN. ]...
   }
}
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You can do this because you have a strict request-response protocol.

If you had a more asynchronous protocol, you still have lots of simpler options than select. These include IPC::Run, Expect and threads.

when I google the errors I get when I don't use it, everyone says to use it :-).

What errors would that be?