You don't mention how long of a fast you're considering, but I assume it won't be months. In that case, the raw greens shouldn't pose a huge problem, but to be on the safe side, keeping the cruciferous family to a minimum would be a good idea. Also, since it's winter, why not consider cooking some of those veggies into a warming broth, as well as making raw juices? This will help reduce the problem and will help keep you warmer.
The fermented veggies, in my opinion, are fine. They're eaten in small doses as condiments, not as entire vegetable courses, and the nutrition they provide is so amazing that what little goitrogenic properties they possess after fermentation (a little, I believe) isn't enough to diminish their health-conferring properties.
Also, most hypothyroidism is due to Hashimoto's thyroiditis, which is about the immune system and not so much about its victim, the thyroid gland. Not that you'd want to over-consume goitrogenic vegetables in this case, but it reduces the importance of this aspect.