All the OpenGL picking tutorials I have seen involve creating a picking render pass when the mouse button is pressed.

It would be much better to do this in one pass, is it possible?

From what I understand, the answer is no. However you can render low resolution models (or just boxes) during the picking pass to speed it up a lot.
The best way to do picking is to implement it on your own using bounding boxes on the models (so it doesn't require rendering at all).

Yes thats what I am doing, it just seems very wasteful when your already processing the full model anyway. Any idea if Direct3d has this same limitation?