while (<FILE>) {
    my $var = scalar $_;
    $response->body($var);
    return $response->finalize;
}
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$_ is already a scalar. Calling it in a scalar context doesn't change anything, so you can also use

my $var = $_;
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If you return from a while loop, the loop is done. In fact, the whole subroutine is done. The next iteration never happens.

autoflush is documented in IO::Handle:

Any unread data in the buffer will be discarded

Are you sure you want to do that?

Update: You used flush on the input, not output. But perlvar for $| which is the same as autoflush says:

This has no effect on input buffering