No welding... Welding is not an option. First, there is the heat it will destroy the ski. Second, you cannot ground a tiny piece of segemted edge. Third, the edge will be even more brittle after welding. Fourth, the base would totally contaminate the fusion bath. Forget it you cannot weld edges and it would be easier the rip them out and apply new ones. But don't do this either because there is no guarrantee that they will stay in for a long period of time.
You are a victim of the steel rail....just keep abusing your skis to the upmost degree. If you are lucky enough that the edges haven't ripped out yet just grind the whole edge so none of the segments catch and move on...my condolences