Have you ever looked at the EDT article?
I'm not telling you how you should do this, but this how I would do it if I wanted to lose a lot of fat fast.
I would do the 5/3/1 workouts 3-4 days a week depending on my workload, stress, etc. I would do some of those EDT style circuits for the rest of the workout. I have a naturally fast metabolism and my body doesn't like to hold on to fat or muscle so I would do some sort of intense cardio 30minutes 2x a week (like the stuff you mentioned).
I have't incorporated EDT into 5/3/1, but I have done it on it's own and it turned my body into a lean fat burning furnace and was the only time I had a clearly visible 6-pack.