TBH it is time to stop analysing and I am going to be blunt and stop pussyfooting around.
You know what you need to do. Archie cannot stay with you whatever.
So, there are 2 options:
1. You have a potential rescue space for him with a place that is experienced with his type of issue. They are far more likely to be able to turn round his behaviour than you are. Yes, it is a long way and difficult to get him there but, if you can't do it yourself, you have plenty of suggestions of how it could be done. If you want to give him every chance of a life at all, you need to get him down there asap, no excuses! There are people on here who have turned round dogs not dissimilar to him and in a different, child free, household the dogs have been perfectly fine.
2. You have him PTS - problem solved, although it is not a great outcome for you or Archie.
His (and your) quality of life sounds pretty grim at the moment and so this situation cannot continue.
There is no magic solution to this, you have tried, it is not working, so you need to make a decision and go through with it!!