I think spring (Sept-Nov) and fall (March-May) are the best times to see Chile by bicycle. The summer, though dry, can be too hot, especially up north (though I had a very successful January camping trip in the south). And the winter is too rainy. In poarticular, September is one of my favorite months. Not only would you be there for Fiestas Patrias, but the weather is just particularly nice.

Spring, with no doubts. Summer is too hot, from north to the center. In the south of the country it rains no matter what station (of course it doesn’t rain a lot in spring-summer, but is not as hot as in the north), but you won’t have problems. September and October are the best months, and in september you have the independece day too!