Minimally tested:

use strict;
use File::Basename;

errlog_on();
warn "$0 started ". localtime() . $/;

warn;

errlog_rotate();

warn;

# ...

{
  my ( $errlog_file, $orig_stderr );
  INIT {
    our ( $prog, $path ) = fileparse( $0, '.pl' );
    $errlog_file = "${path}${prog}.err";
  }

  sub errlog_on {
    $orig_stderr and
      defined fileno $orig_stderr or
        open $orig_stderr, '>&', STDERR or die "Failed to dup STDERR";
    open STDERR, '>>', $errlog_file or die "invisible error";
  }

  sub errlog_off {
    return unless defined fileno STDERR;
    defined fileno $orig_stderr or die "Can't find original STDERR\n";
    close STDERR or die "failed to close $errlog_file";
    open  STDERR, '>&', $orig_stderr or die "Can't restore STDERR\n";
    close $orig_stderr or die "failed to close STDERR dup\n";
  }

  sub errlog_rotate {
    errlog_off();
    log_rotate( $errlog_file ); # backs up $errlog_file
                                # and truncates it; dies
                                # on failure
    errlog_on();
  }
}
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