I read the FileHandleFileCache documentation differently than you do. I think that you're basically supposed to replace your calls to open by calls to cacheout, that is, instead of open $file{$key}{fh}, ..., use:

$file{$key}{fh} = cacheout $file{$key}{name}
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But I haven't tested that. $path is the (path and) name of the output file, and $mode is the file mode (which is irrelevant to your needs).

Update: kyle spotted a link to the wrong documentation.