I was wondering if it was faster to render a single quad the size of the window with a texture the size of a window than to draw the bitmap directly to the window using double buffering coupled with the platform specific way of drawing to a window.
Using textures is usually a lot faster, since most native drawing APIs aren't hardware accelerated.
It will very probably depend on the graphics card and driver.
The initial setup for textures tends to be relatively slow, but once that's done the drawing is quite fast -- in a typical case where graphics memory is available, it'll upload the texture to the memory on the graphics cards during initial setup, and after that, all the drawing will happen from there. At the same time, that initial upload will also typically include full a full mipmap down to 1x1 resolution, so you're uploading a bit more than just the full-resolution texture.
With platform specific drawing, you usually don't have quite as much work up-front. If only part of the bitmap is visible, only the visible part will be uploaded. If the bitmap is going to be scaled, it'll typically scale it on the CPU and send it to the card at the current scale (and never upload anything resembling a mipmap). OTOH, virtually every time something needs to be redrawn, it'll end up re-sending the bitmap data for the newly exposed area. It doesn't take much of that to lose the (often minor anyway) advantage of minimizing what was sent to start with.