May be re threaded Italian, unless it already is Italian threaded.
if so it explains right cup coming loose, its not LH threaded..

might look for a threadless BB.. some are made..

a kludge, lay in molten brass over the threads, then there is metal to re thread again.

but heating to molten brass, may loosen the brazing holding the shell on the frame tubes.

you might look for a shiny new frameset, this time torque the bearings down to spec.