If the weekly task is relatively isolated, start another POE::Session to do it. Something like this may be sufficient if do_that_thing() doesn't take too long.

POE::Session->create(
  inline_states => {
    _start => sub {
      my $next_unix_time = calculate_when();
      $_[KERNEL]->alarm(weekly_task => $next_unix_time);
    },
    weekly_task => {
      do_that_thing();
      my $next_unix_time = calculate_when();
      $_[KERNEL]->alarm(weekly_task => $next_unix_time);
    },
  },
);
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If do_that_thing() takes too long, then it may need to involve system(1,"command"). See perlport for discussion of system(1,"command") on MSWin32. For best results, use POE::Kernel's sig_child() to detect when the command has ended.

And a word about fork() in POE on Windows. If you plan to do anything more than system(1,"command") in the "subprocess", then you should check out POE's Github repository for pre-release iThread-safety fixes. As you may be aware, fork() on MSWin32 is implemented in terms of iThreads, so modules must be iThread-safe to fork() there.