In the past I have experimented with mini-doses of injected glucagon to turn around a low. There have been some formal studies done on this but I don't have them at my fingertips. I stopped doing this because the current commercially available glucagon is a powder that must be mixed with a provided liquid. This solution, as I understand, is only potent for a few days.
As others have already said, the root of your problem is the current background, or basal, insulin that your son takes. A perfect basal pattern, whether delivered via a pump or syringe, will match the glucose output of the liver only, no more, no less.
One way that I've temporarily solved this problem in the past is a bedtime snack of peanut butter. It has protein and fat and digests slowly. I would try to use peanut butter that has no added sugar.
Good luck. My heart goes out to you. I couldn't imagine managing the blood glucose of another person! I would recommend that you look for forums that target parents of young children with T1D. There are some here but some communities are almost exclusively parents of T1Ds.