Could one day of taking rhodiola cause a big drop in thyroid levels?

No. There is no evidence that rhodiola would reduce thyroid levels whether taken for a day or a year. Rhodiola is thought to generally stimulate thyroid function rather than impair it.
Very unlikely. Rhodiola is a adaptogenic herbal remedy and there is no evidence that it would have an effect on thyroid function. A big drop in the thyroid level is most likely caused by a problem in the thyroid.