This would mean the controller would lose positive identification of which aeroplane is which in the situation at a fairly critical time. Dumb idea.
Any time I've seen a 76 or 7700 squawked, the system retained the datablock identification normally. It just went red.
Of course, if it was a coordinated response, and both transponders went to the same code from within the series, identification would be lost.
Since TCAS units are able to coordinate their responses (you climb;I'll descend) it probably isn't beyond the tech abilities to have the climbing one assign itself, say, 7201, and the descending one 7203 (or whatever.)
Of course, not all ATS systems have the same functionality, so it might be a "dumb idea", but I would have thought, in the wake of Uberlingen, one worth at least investigating.
Yes, it does only take a few seconds to notify a TCAS RA to ATC. Provided the frequency isn't jammed etc.
In our neck of the woods, it states pretty clearly in MATS that control instructions are not to be issued to the aircraft reporting the event or to any other aircraft involved. Which
might exclude passing control instructions to a third aircraft, unless it was known for sure not to be involved (yet).