Pick the Party Cannon with 60% accuracy, then deliberately miss your first shot.
First instinct would be to pick the best Party Rocket Launcher for a 100% shot at eliminating one of the competitors, but you'll soon find this strategy wanting. By picking the best weapon, you have made yourself an obvious target for Black Bird, who has a 70% chance of taking Rainbow out: the one shot would naturally be used on the opponent with the best shot: Old Fogey.
So what if Rainbow used the Party Cannon with an 80% success rate? As long as the odds of making the shot aren't the highest, you know Rainbow won't be a target until Old Fogey's out the running, but then Black Bird, again, would have a 70% chance of knocking Rainbow out, assuming she managed to actually knock out Old Fogey. If Rainbow takes out Black Bird, Old Fogey will take out Rainbow afterwards, nine times in ten.
However, what if Rainbow misses her opponent with that same 80% success-rate Party Cannon, INTENTIONALLY or otherwise? Well, then that would mean Black Bird would attack Old Fogey, the better shot, then he'd incapacitate Black Bird, leaving Rainbow free to shoot at Black Bird, and four out of five times she'd succeed...if it wasn't for the fact that Black Bird can catch on to this, deliberately fluff her shot as well, then, as your 80% success rate is higher than Black Bird's 70%, Old Fogey's going to try and take Rainbow out first - remember, he has to try and take someone out now, or else all of them lose.
The answer is therefore clear: take the weakest party Cannon, then miss on purpose. That way, Rainbow is almost guaranteed to survive after one round, then when it's her turn again. If she misses, Black Bird will have to take out the better shot now (Old Fogey) or else Old Fogey will attack back. If she succeeds, Rainbow has a 60% chance to take out Black Bird. If she misses and Old Fogey takes out Black Bird, Rainbow can take out Old Fogey with a 60% success rate, and that's the best she can do.
There IS a 3% everypony will miss and everypony gets gummed, but when all is said and done that's the most optimal strategy.