How would the opponent overguess you? You can delay your strategy switch for as little or as much time as you like, since their best strategy only breaks even.
If they don't know what you're doing, you could rack up more wins by exploiting flaws in their strategy, and rack up even more wins when they "wise up" and try to counter you. I guess the difference between our answers is that I'm treating this as an actual game against another person, whereas you're looking for more a game theoretical optimum.
Now, if your opponent does have perfect knowledge of your strategy, then randomly alternating between the strategies would be the best approach, which is effectively the same as your 1/3 rock, 2/3 paper idea. By the way, I ran some sims in matlab just now, and I'm getting the optimum to be bang on 1/3-2/3, so the math might not be so interesting after all :-(
There are only two degrees of freedom: the probability that we pick paper (assume we pick rock the rest of the time -- picking scissors should never make sense); and the probability that our opponent picks scissors when they have the choice. I'm assuming that our opponent never intentionally chooses rock. That seems fair since our plan involves throwing paper more often than not.
Making a meshgrid of those two probabilities in 1% increments, running 100k games at each point, and then taking the minimum along the axis of our opponent's choice (i.e. assuming they always pick the best strategy), I find a peak at 68% with us having a 16% advantage, which is about 1-in-6. So it looks like your initial guess in the other post was correct.