t3 is sometimes used as an add-on to antidepressants for people who have lingering depression symptoms. it either works or it doesn't and you'd know within about 4-6 weeks. likely there is nothing intrinsically wrong with your thyroid if all your tests are normal but nevertheless boosting the level of thyroid hormone (not "balancing it out" -- quite the contrary; making it run higher) is an age-old strategy for helping to treat depression. i doubt it would do very much in terms of weight. it does not damage your thyroid.
can you explain what t3 will do to help balance out a thyroid that has come back as being normal based on testing?