Referees will get grief for applying the law here and will get grief for not doing so (although in this scenario that is very unlikely). Don't worry about the grief, deal with it as you see fit.
When refereeing a game, my last concern is whoever is refereeing these teams in their next game. I have enough to deal with in my own game.
Use your judgement on your game and the next referee must do the same too.
We are talking about 13 year olds playing in what I assume is a low standard of football (considering the score) and the score is 13-0. I have hardly done any kids football in my 8 years of refereeing, but there is a mercy rule because situations where one team hammers the other has alot of negative effects on the losing team. At 13-0 at this low level of kids footy, I'd suggest your management skills need improving if you're going for the red card for a DOGSO. That wouldn't be a criticism of anyone awarding a red card because they've done so knowing that it won't necessarily be accepted well and it's turned what is probably an easy game at 13-0 into a tough one for the referee. But that is a situation that could be managed better, and managing people/games is important.
I am in no way saying doing this in a game which is competitive. There are also other factors at play like goal difference, maybe another team battling relegation with the team who's 13-0 down etc. This is where your judgement comes in. But they're 13 year old kids and it's been a pretty grim experience for them already. Mercy?
