SOS message sending “Failed”

I still believe it is low power mode. Do you have an SMS/MMS plan for your iPhone? I've never used the SOS, and I've not seen how it sends the messages. This support document, Use Emergency SOS on your iPhone - Apple Support just states it will send a text message, and since emergency services uses cellular, it may not send an iMessage, but a text message (SMS) which requires you to have the SMS plan on your phone. Check with your carrier to see if they can explain why the messages did not send.

It may have to do with the fact that your device is in Low Power mode and almost dead. Do you see any other error when you open the message?