First, the point about changing codes 'on the other transponder' might be worth considering, but most aircraft I have flown have two transponders drivenm by one selector head, so only one code is ever displayed - you just select which box (and which Mode C feed) is active.
Second, it is my understanding that whilst ATC radars do take their time to receive and process SSR transmissions, the same is not true of airborne radar as used in SAR, for example on the British Nimrod, which does (I am told) show an emergency squawk on first reception. Would a Nimrod crew-member care to comment... perhaps someone at LATCC could make a 'phone call to ask the question....