I taught a socket chisel class as well this summer. My students were having a hard time getting the very end of the sockets to bond together. We started with small stock (1/4 x 3/4), drew out a triangle, folded into a cone and welded. My demos welded ok, but students don't move as fast. The narrow end of their sockets welded successfully at the anvil, but the wider section formed over a bickiron did not. I think they just weren't getting onto the bickiron in time with sufficient heat. I have ideas for the next time I run this class and I'd love any thoughts or suggestions you have!
Make bigger chisels, starting with thicker material so it retains heat longer.
Preheat the bickirons. Not to red, just a few hundred degrees to prevent them from sinking as much heat.
Try to set the weld with tongs while it's still in the forge (gas unfortunately). Once the wide end of the cone is stuck I think the rest would go more smoothly.