If you go very low carb you enter ketosis. Generally ketosis and high intensity exercise don't mix well. You can do some, but you'll need to adapt for a few weeks before you'll have the energy.
If you're going to do high intensity exercise, I'd recommend bumping the carbs up a bit. But strictly from fruit and potatoes/yam. They don't irritate digestion and induce leptin resistance like grains do.
If you do want to stick with ketosis, make sure your fat intake is higher on dairy/coconuts. Makes ketones easier than other fats.
Do you understand the theory of evolution? We didn't start eating grains until about 10,000 years ago, and not significantly until a couple of thousands of years ago (and even more starting 40 years ago). If the negative effects of something are weak and don't impact heavily until after you reproduce, it takes a long time to select for.
Like how some people can't handle lactose with a digestive enzyme called lactase, a lot of people can't handle phytates with a digestive enzyme called phytase.
http://en.wikipedia.org/wiki/Phytic_acid#Food_science
Grains also have other evolutionary mechanisms to protect their young against predictors, like lectins, gluten. Look up "intestinal permeability" on a medical database. Grains cause this.
Grains also have shown to induce leptin resistance because if they do make the intestines permeable they also make it through the blood brain barrier and bind to leptin receptors without inducing satiety:
http://www.biomedcentral.com/1472-6823/5/10
However, just because we didn't adapt to something theory with evolution doesn't necessarily mean it's harmful. Potatoes are a good example of this.
http://wholehealthsource.blogspot.com/2010/09/potatoes-and-human-health-part-i.html
http://wholehealthsource.blogspot.com/2010/09/potatoes-and-human-health-part-ii.html
http://wholehealthsource.blogspot.com/2010/10/potatoes-and-human-health-part-iii.html