I can't imagine weight training to ruin your cycling unless you train in bad form. I discovered that squatting and deadlifting has given me more options when climbing inclines on a bike. Instead of the usual push with your quads peddling, I can opt to stick out my butt a bit and push with your glutes. You can feel your hamstrings working too.

Also, I believe it's a myth that weight training isn't cardio exercising. Have you ever felt your heart pumping and your breathing harder as you push or pull with low reps/high weight or low weights/high reps? Doing exercises right after another with little rest and you would have put in probably equivalent to 20 minutes worth of straight cardio.