You sound pretty busy. If you are new to cleansing, I would see how it goes. A few years ago, i experimented with going to the gym just to see how i would do while fasting.
It went well. Now i work out 3-4 days per week while fasting. I sweat extensively, feel more relaxed and it really does assist with the cleansing by burning up more caloreis and fat.
I have been fasting now 30 days per year for the past 5 or 6 years.
That may work, the only way to really know is to try it. Just take plenty of lemonade. I would be extra carefull if you will doing alot of inversions and forward bends.