Excellent question. For almost 50 years we have known that glucagon is high in DM1, even more than in DM2 because there is little effect of paracrine insulin action on the alpha cells due to the beta cell destruction. We may use Symlin (pramlintide) an Amylin analog for this purpuse (that is approved by FDA in USA). But the same may apply for GLP-1 analogs or the incretin enhancers the dipeptyl-peptidase-4 inhibitors (DPP-4-I).
Roger Unger, an authority on glucagon (and many other things related to diabetes) has mentioned decades ago to employ suppression of glucagon as a therapeutic measure.