Ballast checks from your phone. To hear about a dead server right away, something off the phone has to notice.
You run these on your server. Ballast doesn't run them and can't: it only reads.
Ballast has no server, and iOS decides when it wakes. Sending to ntfy or Bark from the phone wouldn't help, because that would only happen when the phone wakes as well.
Have your backup ping a dead-man service after each run. When the ping doesn't arrive, the service alerts you, even if your server is dead.
Add this to your backup script, with your own command and ping URL. Ballast fills in the URL for you under Settings › Alerts while your phone is asleep.
your-backup-command && curl -fsS -m 10 --retry 3 https://hc-ping.com/your-check-id || curl -fsS -m 10 https://hc-ping.com/your-check-id/fail
The /fail part is for Healthchecks, and tells it right away that the run failed. For Uptime Kuma, stop before the ||.
Point Healthchecks at ntfy, or call your service from the same script. If you host ntfy yourself, instant delivery to an iPhone needs its upstream relay set up: see its configuration docs.