I regularly throw 1/0 and larger deer hair/foam bugs with my 4 weight into headwinds. I can also chunk those heavy streamers with the giant dumbbell heads a pretty good distance too; it just comes down to taking it very carefully with such a light rod. Double hauling is pretty much out of the question, as it seems to put too much stress on the rod. The trick I use is to sidearm with the fly as low to the water as I can keep it.

I only do this because I generally fish large areas for gills, but I'll come on a spot that looks like prime bass area, or I'll see bass rolling on the surface and quickly tie on the fly.

The biggest problem is getting the fish in, rather than casting. Good luck keeping a 4lb bass out of structure with a 4wt and 5x tippet, lol.