Thankfully, I didn't have to "get" him to do it. He knows that suppressing a Hashi-damaged gland completely and then replacing with Armour is often the best way to eliminate symptoms.
Doctors are most often worried about osteoporosis due to overtreatment, I think. It's a concern that's misguided, but stubbornly persists. My MD sent me for a baseline DEXA scan, and he stresses I should take calcium and minerals for bone health, but I don't think he's super-concerned about the thyroid med causing damage. He's very smart...
The only thing I've been able to find about suppressing TSH online is Dr. Anthony Toft's paper on the treatment of hypoT. He is a consultant physician at the Endocrine Clinic of the Royal Infirmary in Edinburgh. You should be able to find his writing pretty easily. He says that most people do best with a dose that results in high-normal FT4 and low-normal TSH, but that some patients achieve a sense of well-being only if TSH is low or undetectable and the free T4 is slightly elevated.
Wish I had more to offer you. Hope that helps.