The cleanse would probably make things worse. Fasting slows your metabolism, which slows digestion and leads to constipation. Your body also needs a constant supply of glucose, and in the absence of carbs will break down lean mass to produce it, with ammonia as a by-product. If you’ve ever kissed anyone on the Atkins diet, you’ll probably be familiar with ammonia. Eat to preserve health-promoting lean mass, and try checking your basal temperature. If it’s low, then raising it could solve the constipation if it isn’t a problem you had before.
Extreme changes in diet are a shock to the system, so side-effects should be expected. What you want is stability. Your body has a lot of adjusting to do if you’ve changed your macronutrient ratios and the content of your meals. More fibre and saturated fat require more bile and beneficial bacteria. These adaptations don’t happen overnight, so start trying to get some consistency in your eating habits.