Catalyst stderr going to the wrong log

I have several vhosts served by Apache 2.2.22 on a Debian Wheezy server. Some vhosts are Catalyst apps; all vhosts run with mod_fastcgi and mod_suexec. Each virtual host is configured to use a different ErrorLog.

When code in the Catalyst apps writes to stderr (e.g. warn is called), the message ends up in the main Apache error log instead of the one specified in the <VirtualHost> block. It's not prefixed by a timestamp or any other information.

However, if I call this trivial fastcgi script

#!/usr/bin/perl

use FCGI;

my $request = FCGI::Request();

while($request->Accept() >= 0) {
    print("Content-type: text/html\r\n\r\n");
    warn 'Hello!';
}

the Hello! does end up in the right place, correctly prefixed:

[Thu Jul 10 14:40:18 2014] [error] [client x.x.x.x] FastCGI: server "/data/vhost/wibble/docs/test.cgi" stderr: Hello! at test.cgi line 9., referer: http://my.test.site/

How can I get the Catalyst sites to write their stderr to the correct log?

The only related question I can find is this one -- similar problem but different module (mod_perl), and sadly no answers. I also found this Apache bug report, which describes the same problem with mod_cgid.

Edit: If I put

open STDERR, ">>../../applogs/app.log" or die "can't open applog: $!";

before the Catalyst loop, this redirects stderr to my log file, but obviously no datestamps or other prefixes are added. What I want is for stderr to be treated the same as for a bare FCGI script.