I don't understand how a low-carb diet could encourage fungal growth. Especially when (Wahls) the diet emphasizes vitamin D supplementation to assist immunity, as well as several other nutritional modifications to bolster immune function.
Thyroid function can be disrupted if you eat too many raw cruciferous vegetables (as mentioned above) but you can supplement with iodine-containing foods (kelp, for instance).
I don't understand what is in bone broth that would cause flushing. From what I understand, you would start a very restricted diet (no dairy, no gluten, only approved carbs such as squash, some fruit, etc.) and have your bone broth everyday as well as a range of veggies while using probiotic foods (or supplementation if you can't handle basic probiotic kraut) and do that for several weeks to allow your gut to heal. Of course, if you're taking antibiotics concurrently with this, that throws everything off.
I don't understand how fermented foods (for instance, cabbage soaked in salt water for 7 days in an anaerobic state) would cause flushing. Obviously, I don't know your specific health concerns and there may be very good reasons for why these are triggers for you.
I haven't heard of people developing idiopathic flushing from following the Wahls protocol, but then again, I haven't been looking for examples of this--I've just been trying to adhere to it as closely as I can. And I feel better for doing it.
Not sure that this helps much, but for what it's worth.