I've seen them, old Carloader with no water pump.
The one of them I worked on had a J&S propane regulator mounted in a special radiator hose outlet that bolts on the head or it fit in the hose itself. That would require a different carburetor,though. Is there 2 bolts that hold what looks like a thermostat housing on the head where the hose hooks on? That might be the only way to go unless you can drill and tap for fittings in that hose housing and maybe find a drain plug in the block to use as a fitting hole. But the water flow may not be good enough. Your regulator probably needs a rebuild and that is causing it to frost up real quick due to running so rich.
I don't think the broken cap is causing any problem.