"What is the correct procedure?"
I'm not sure, but a word of caution about firing a primed empty case in any double: without the normal pressure of firing, the pin may stick in the primer and make the gun difficult or impossible to open without damaging a firing-pin. If this happens, it will be necessary to drop the floor-plate and manually recock the tumblers in order to open the gun. Graeme Wright mentions having to do this in his book.
Regarding proof-firing, I would probably load a round in the first chamber and fire it, then check. This may not be the correct proof-house procedure, but that way if something is distorting you may catch it before completely ruining the gun.
Thinking about it, a rigorous procedure might be to fire a proof load in the first barrel, then check everything. Repeat in the second barrel, and check again. If all is well, then fire a left-and-right, a couple of seconds apart, then a final check. If it passes that, you definitely have no worries IMHO!
