I did an extended induction period of Atkins when I first started, but I gave my body time to adapt to the diet changes before doing hardcore workouts.
On one side of things, it sounds like you are way more active than I was at the time.
On the other side, you may need to get past the “induction flu” before going back hardcore.
I now do heavy lifting 5 days a week on a fairly low carb plan and have more than enough energy to make it through the workout and beyond

I tend to spread my carbs out across the day and use fat as my fuel source. If you want a few carbs before working out, you might try some berries.