You could take them to a welding shop and get them welded or brazed for 15$-25$ max. Small spot repair like that would take a professional only 10-15 minutes.
Epoxy glues like JB produce toxic fumes when they burn, (like most glues and plastics.) I believe it says right on the package "not for high-temperature use." More importantly, after it burned and charred, far from fixing your problem you would have simply created a bigger mess.