I have had some similar issues with my five-year-old son, not so much avoiding a challenge but avoiding taking it on in front of others. Some of the strategies I tried:
* explaining to him that practice is often needed to achieve mastery, that needing practice is no shame, etc. etc. etc. (I know some of these messages seem to fall on deaf ears with a perfectionist child, but I still believe that they have an effect over time)
* letting him practice on his own, with the idea that he must periodically be open to doing it in front of us
* finding activities that inherently encourage or incorporate imperfection during intermediate stages, with the perfectionism allowed to take hold in the finished product (i.e. building mockups of a final model)
* encouraging him to do something with me at the same time, so that we collaborate-- or seem to
As an example of this last one, for a while he was stressing out about doing any reading in front of us, if there was any possibility of being corrected when making even a quite minor mistake. I decided to try reading with him. We picked out some Disney duck full-length comics and began reading them together, pronouncing each word at almost the same time (I actually, once we got going, started lagging a tiny fraction of a second behind). When he caught on to what I was doing, I would read at the same time, then start lagging ever so slightly again. He eventually simply kept quiet about it, though he knew what I was doing, and soon was able to read in front of me again, though he still likes reading out loud together.
I think that collaboration is powerful because it can lessen the perceived harm from a mistake, because faults can be shared, and it presents chances for a child to be corrected in passing without it being a big deal. So, for example, even if I'm lagging behind on a certain word, and he mispronounces it slightly, he can note my correct pronunciation and drive on without it being a big deal (although once the technique started working, he started re-saying mispronounced words without a hassle).
