Actually, I think I just figured it out.

if ($Win32::Daemon::VERSION < 20180000) {
    Win32::Daemon::AcceptedControls(SERVICE_ACCEPT_STOP
                                  | SERVICE_ACCEPT_PAUSE_CONTINUE
                                  | SERVICE_ACCEPT_SHUTDOWN
                                  | SERVICE_ACCEPT_PARAMCHANGE
                                  | SERVICE_ACCEPT_NETBINDCHANGE);
}
[download]

The problem seems to be that this should not be in an if statement. For some reason I thought this call did not need to be made for Win32::Daemon version 20181025, which is on this system.