I say the scout treating him self is fine. Like Brian says many youth have more experience and training in 1st aid than the adults. Make it a point to notify SE and Camp Director. Also tell the scouts that they should always let an adult leader or SPL know about injuries even if the were self treated no matter how small they are
I've never had a Scout who got to the point of crying for hours, but it could easily have been my oldest son. He is 18 now and earned his Eagle rank last fall. He has dealt with severe social anxiety for many years. He started Scouts as a Tiger Cub in 1st grade, so he has grown up in Scouts and camping. However, his anxiety is one of the reasons that I got started as a leader 12 years ago, actually being the Committee Chair for 10 years and now 2 years as Scoutmaster. My being there eased his anxiety to where he can function. He learned a long time ago that I'm not there as dad, I'm there as Scout leader. I didn't want to bring any dad/ son issues from home to Scouts, so I stayed out of their way to let the Scouts lead & function as a Patrol. But by me being there, he was able to function and be a part of the Troop without having his anxiety/ breakdowns. Is there any chance that you or his dad could be active, at least as another adult on the outings or to spend some time at camp? I know there are several who will say to leave him and let him get past the anxiety on his own. But having a son who can be paralyzed by his anxiety, I have become much more sympathetic to the youth who have very real issues. Many of these issues will not be overcome by just "sucking it up and dealing with it". However, there is certainly a time for some Scouts to just suck it up and get over it. As the Scout leader, it isn't always easy to discern which Scouts need which level of response.
As far as making him do things that trigger his anxiety, I will add that from the time he was young, we have been easing him into uncomfortable situations. Things like making him start to order his own food at a restaurant. This forced him to look at a menu and find what he wanted. It forced him to speak clearly and loud enough for the cashier/ server to hear his order. I would also let him answer questions about his order (i.e. do you need any sauces?). There have been many things like this that I have pushed him into, some of which have been very hard for him. But over the years, we look back and see that each step has moved him towards dealing with his anxieties. It's much easier to deal with that level of anxiety knowing someone who knows you is nearby.
I say all of that to say that you know your Scout the best. I hate for his anxiety to keep him from all that Scouting has to offer. Scouting can offer him ways to face and deal with his anxiety. Look for ways that you can help him to deal with it & support him, while maintaining some distance and give him room to excel and thrive, knowing that he isn't having to do it alone. It may mean that you or dad end up on Troop campouts. There are certainly worse things than that.