Since you're having these symptoms on only 2 drops of iodine, that's a sensitive situation. Here is the complete list of recommended supplements to help detox bromine.
From Citingsources (Lynne Farrow, Breast Cancer Choices) Bromide Detox Strategies –
1. Salt load and drink lots of water with it. Add at least ½ tsp unprocessed salt to daily diet. This can be in the form of ¼ tsp unprocessed salt per quart/liter of water.
2. Pulse-dose iodine. Stop Iodine for 48 hours to rest the kidneys while continuing Companion Nutrients (see below).
3. Reduce the Iodine dose temporarily, then work back up.
4. Take Vitamin C 3 grams or more spread throughout the day.
5. Take Iodine Companion Nutrients to enhance cell detoxification –
ATP Cofactors 2 a day (each tablet = B2 riboflavin 100 mg + B3 niacin 500 mg as inositol hexanicotinate)
Magnesium 300-600 mg
Selenomethionine 200 mcg
6. Use Zinc 25 mg for skin symptoms.
7. My addition – Oil Pull several times a day, at least 2 hours apart, using 1-2 tbsp organic sesame oil. Swish for 5-20 minutes. If this works for you, when you spit out the oil it may be quite stinky.
Here are two discussions from Stop the Thyroid Madness about Hashimoto's and one person's experience with thyroid nodules. Both involve using iodine.