Can you give more details about how the zip file is created? Does it just consist of files that already exist in the filesystem? Or does your code need to create the contents on the fly by carrying out expensive calculations or retrieving data from another source?
The key question is how much of the 10 minutes is taken up actually doing the file I/O needed to write to $file?
If the answer is that the 10 minutes is accounted for by file I/O, then
a possible approach is to use a combination of HTTP chunking and streaming the zip file direct to the client as you create it.
Info-Zip can create a zip file in streaming mode, as can IO::Compress::Zip.
Don't know Apache2::Request at all, so I can't comment on whether it supports chunking.