Yes, you can take Iodine while you are on thyroid meds if you are willing and able ti monitor your symptoms and adjust the dose of meds when necessary.
Having said that, a drop of Lugol's does not have 150 mg, it has 6.5mg; therefore I suspect you have a kelp (or other) product with 150mcg of iodine. It takes 1000 mcg to make one mg. So you need not be concerned about overdosing. When I took kelp tablets, I took 5-6 per day and if it improved my health, I did not notice. I got off my meds, not thyroid, after three months on Lugol's.
Use the user finder at the top of the page, and look for Prism's posts. She got off thyroid meds using Lugol's. I also agree with the suggestion to read all the FAQs under both forums. Read, read, read!!!