You've had so much good advice and experienced wisdom in this thread. I'm not going to vote one way or another. I did want to address a couple of points you made though.
1. Game roosters are often the friendliest to humans, just not always to other roosters. They have been bred to fight other males (sometimes just distant genes) not humans. A bit like pit bulls. Human friendly, not dog friendly.
2. Don't underestimate the assholeness that teenage hormones give a rooster. Most, but not all, settle down and behave sensibly later on when this phase has passed. If you wanted to see if this was the case, you'd need to separate and enclose him for a few months. I understand you don't have a separate coop and run to do this. It's not difficult or expensive to construct a small temporary coop and run, which might come in handy at any time when you need to separate a chicken, male or female, for bullying, illness, or new ones prior to integration. In fact it's a good idea to have one anyway.
3. Your children have voted to cull. Their relief at feeling and being relatively safe when he is gone is very valuable. I reckon you will all feel relieved, not to mention their enjoyment of your outside space and your peace of mind.
4. The dog thing I would worry about. If your dogs start to defend you against chickens, they may well overreact and start to attack the others. Then your trust and your flock are under threat.
5. You say he is good looking which is why you want to keep him. Sorry to be straight talking with you, but to get some perspective, he is nothing special to look at. A run of the mill barnyard mix with common male colouring. Sorry. I mean no offence. Millions of other roosters are better looking. <runs and hides!> I bet your Faverolles are way more handsome!
6. And this brings me to my final point. It's not like he is your only rooster, and you are heavily emotionally invested in him as a pet. You have many others to select and keep, and you already have observed good behaviour in the others. This should make removing the human aggressive one an easy decision.
7. A second final point! As to method of removal, you have choices. You could cull and soup him, or rehome. I've never culled an unwanted healthy rooster, and trust me I have had so many over the years. Somehow I have always managed to find new homes, whether with breeders, pet keepers or possibly soup makers. Sometimes it takes a few months and I have to manage them in the meantime. You may want instant relief, which is your decision and fine.
8. Ok, a third final point
. This is your family, your children, your dogs, your farm/household, your chickens. You must manage them as you see fit. Nobody on the internet can tell you want to do or make your decisions. There can be no judgement on what is best for one person which might not be best for another.
I wish you all the best in whatever you decide.
