Could an airliner exceed Mach 1 in a zero-G power dive and safely recover?
There is only one answer here and that is NO, especially for the A320 in your example (there are other airliners better suited to tolerate higher transonic speeds).
Yes, it's possible to recover from such a condition, but nothing about it would be safe. Recovering from this condition would be a situation in which you would consider yourself lucky, because the aircraft is simply not designed to operate under such conditions and catastrophic mechanical failure becomes a serious possibility. Successful escape will always be on account of some degree of good fortune (ie: things you didn't plan, didn't think to plan, or couldn't plan, but that went OK and conspired to save your bacon). The aircraft will likely suffer damage - with luck, it isn't critical damage.
Who is to say what the weak link would be, but it could be anything - a single mechanical component could mean the difference between life and death. Does your plane have a critical support that's on the weak side of the bell curve? Maybe a sister craft made the next day might survive a dive that yours would not. Each plane will fail in a different way depending on whatever component happens to fail first. When you're outside the envelope there are no guarantees - you might get lucky, you might not.
How do I get out of this in one piece?
What's important to understand is that the outcome in these situations becomes highly unpredictable. Subtle changes in the test conditions, weather, environment, etc, can all cascade quickly into an uncontrollable situation, regardless of the pilot's skill. Despite your best efforts, the result may still be fatal.
That someone managed to do it successfully once is not an indicator that it would be safe to attempt again.
how much faster can I go before I need to start this process of getting out in one piece?
It's already too late. The faster you go, the higher the chance you won't make it.
Supersonic aircraft are specifically designed to minimize the destructiveness of the pressure waves they generate and otherwise stiffen the airframe against the residual stresses that result.
Airliners, by contrast, optimize entirely for sub/transonic cruise speed and, critically, efficiency. They incorporate no such design features and in fact have efficiency-minded elements of their design that make them particularly ill-suited for supersonic flight. As such, they tend to generate more disruptive and destructive pressure waves than supersonic aircraft during transonic flight.
Because supersonic stresses appear in largely different areas of the aircraft than subsonic stresses, airliners are also far less structurally capable of withstanding them. Doubly so these days because the aircraft all have a variety of overspeed protection systems which allow the designers to certify the aircraft closer to the danger zone of the flight envelope.
From : IATA -- Loss of Control In-flight (LOC-I) Prevention : Beyond the Control of Pilots
4.8.4 Benefit of Overspeed Protection Systems
Automated overspeed protection systems may be seen as something designed to help avoid a dangerous high speed condition. At least that is the perception of many pilots – why else would a designer make the effort to develop and certify the system if not for the benefit of protecting the aircraft? Well, there is one big commercial aspect which is not so well known, and that is significant aircraft weight savings. By installing an overspeed protection system the designer is permitted by the standards to bring VMO/MMO much closer to the design dive speed[emphasis mine] and hence the speed at which the airframe will eventually start to flutter and disintegrate. In other words, the structural integrity margins between the maximum normal operating speeds used by the pilot and the speed at which damage will occur may be narrower, meaning that the aircraft structure need not be as robust. Structural strength usually equates to weight in aircraft construction so ultimately the overspeed protection system allows a lighter overall aircraft structure.
Airline travel is such big business that the rules bend to allow them to shave efficiency right up to the margin.
For a relatively recent real example of it not ending well : Adam Air 574 : 737-400
The aircraft reached 490 knots (910 km/h) at the end of the recording, in excess of the aircraft's maximum operating speed (400 knots (740 km/h; 460 mph)). The descent rate varied during the fatal dive, with a maximum recorded value of 53,760 feet per minute, roughly (531 knots (983 km/h; 611 mph)). The tailplane suffered a structural failure twenty seconds prior to the end of the recording,:52 at which time the investigators concluded the aircraft was in a "critically unrecoverable state". Both flight recorders ceased to function when the 737 broke up in mid-air at 9,000 feet above sea level.
The cause was determined to be pilot error.