Wow. It sounds like all the adults in this scenario are putting their energies into being right, not solving the problem. Your DC has a problem with homework and he is still quite young. he needs help. No point in people saying 'He shouldn't'. He does. No shame in needing help at times. Don't we all.
Ditch the feud with the school. FWIW, I think their inflexibility isn't helping. As pastoral carers themselves, they must see that you can't renege on your responsibility to your scout troupe and that their suggestions of public transport aren't a viable solution.But he has to do the detention, because he knew the score. And he has to do the extra hour in recognition of how his behaviour is disrupting so many lives, not just his own.
That aside, I think you need to help him set up structures for his homework. If you get heavily involved for a couple of weeks, that's probably all he needs to get him on the right road.
Sort him out with a regular time and place. Somewhere quiet.
Sit with him, get his homework diary out and go through it.
Ask what's needed in by tomorrow, and what is for later in the week. Suggest he prioritises stuff due in sooner, and get him to choose which piece to do.
Compliment him every time he makes a right decision.
Ask which pieces he understands best and see if he can quickly cross off a couple.
If he has vocab, test him on it.
If he has online research, set a timer for him to print out a few sheets and work on them, so he doesn't spend an hour on you tube instead not that my DC have ever done such a thing
If he doesn't understand something, get him to text a friend and see if they can advise.
If he has an essay, ask him to tell you a bit about what he knows about the subject already and then suggest he starts making notes on what he's told you.
Can DS1 help at all - with maths or vocab testing?
Can you agree a reward he gets afterwards for each piece completed? My DC get to watch programmes they're not usually allowed. The argument is that if they are mature enough to do such good homework, on time and without fuss, then they're mature enough to watch Bad Education or something similar.
Do you show interest in pieces he's done? I nab DC's books sometimes to see what score they got for essays I think they did well, or for maths they tore their hair out over, and discuss it with them. That's always a good time to sneak in a 'See - when you work hard you get good results' or a 'stick at it and your grade will improve.'
DS1 went from bottom of class 4/20 in one test to full marks in a couple of weeks because once the tantrums were over 'wish I'd never chosen this subject' he got on with it. But he needed help.