I’m planning on doing this breakthough workout. Should I ride until I can’t keep up with the required wattage, or should I continue riding after that point of failure with whatever wattage I can sustain?
IMO/IME all fitness tests should be ridden in Slope mode where you can exceed targets and durations (if able) and ride to complete failure on any interval(s) designed for that purpose.
If your signature is already dialed in you could ride Version 3 in AUTO mode under ERG control and you should fail on the final interval in each set. But if you finish and think “I could have gone harder/longer” then your sig wasn’t dialed in. 
In Slope mode you can crush both sets to failure with whatever it takes.
I forgot to mention I will be doing the test outdoors.
I’m pretty sure I will fail to sustain the required wattage at some point, but my question is whether I should keep on riding at a lower wattage after that moment, or can I just stop the workout?
As long as you can produce something near a potential new TP it will be useful to keep pushing.
When you ride to complete failure on those intervals, you’ll be coasting for a bit afterwards. 
Set 1 is ~2:30 min intervals @ just above TP and rising through the final burn.
Set 2 is ~30 sec intervals @ higher watts, higher count, similar pattern.
The goal of any BT attempt is to draw down MPA and flatline it for 5-7 sec minimum.
Or you might attempt to punch through your PP if a workout includes some max sprint efforts like Version 2 does.
IRL this means any HIT workout or hard ride has the potential to induce a BT if you decide to go for one. I.e., a maximal effort under fatigue that exceeds your current signature values.
I’m liking the way you can gamify the process by leveraging XMB.
No “formal test” is required to feed Xert’s signature model. 
In short, to get a breakthrough, you will either need to push above your MPA with a hard effort or sustain a relatively steady power output and pull your MPA down to your power output.
In either case, you shouldn’t stop as soon as you reach MPA, but keep going until you can’t pedal anymore! Easier said than done, I know 