My totally non-expert opinion? This sort of advice is usually offered in reference to traditional "body building" routines and dietary recommendations, not low-carb specific.

If you are "fat-adapted" via your low-carb routine, you'll have plenty of energy to do the workout at any time, fasted or not. In fact, there might be some fat-burning advantage to doing it in the morning in your fasted stated. As long as you don't feel dizzy or shaky (beyond the usual challenges), you should be fine. Physiologically speaking, your body is probably not "depositing" protein in the muscles while you exercise. That's an ongoing rebuild, not related to the immediate consumption of protein. Drink water. Coffee with cream is great, too.

Try it out, see how it goes, and let us know. I'm hugely in favor of weight training, and I agree that first thing in the morning is the time to get it done.