A thyroid is typically not removed because of goiter. It's, typically, only removed if you have a nodule that's cancerous or if there's some other issue that necessitates removal. It sounds like your thyroid is swollen and may have some inflammation from the Hashimoto's. A cool cloth on the neck can help reduce swelling and help ease the feeling of tightness. Don't rub or palpate your thyroid area as that irritates it and makes it worse.
Often, once thyroid hormone levels are where you need them to be, the swelling/inflammation will go away. What are your actual thyroid hormone (Free T4 and Free T3) levels with reference ranges? NatureThroid is dosed in mg vs mcg. Do you mean you're on 80 mg NatureThroid? Do you take your medication all at once or do you split the dosage into 2 or more doses throughout the day?
Thyroids are usually removed either because it has thyroid cancer or might have cancer, or if it is causing significant discomfort or pain, or if it is impeding air flow in the trachea or preventing swallowing in the esophagus.
It sounds like you fall in to a pain or discomfort category, and that may be a "gray area" where there might not be a compression of the trachea or esophagus need, but it is causing difficulties in your daily life. It might actually be pressing on your esophagus or trachea - if it is a constant tightness - this would be something that you would need to discuss with your doctor. No one can specify a "size for removal" because we all have different necks and thyroids, and some normal sized thyroids might have a nodule that compresses a trachea, some extremely large thyroids might not be causing any problems.
If you are in significant pain and medication does not help with the symptoms or decrease the swelling, maybe it is time to discuss surgery with your doctor, or at least get some professional medical advice.
As someone who has had her thyroid removed for two years now (after multinodular goiter from Hashimoto's, as well as thyroid cancer in three of many, many nodules), I can tell you that if I could have kept my thyroid and still had some normal thyroid hormone produced by my own body, even if that was insufficient for my needs and needed to be supplemented, I would have. Getting to the right amount of thyroid hormone, and even once I'm "within range", it has taken years to get my energy levels back to where I was before the surgery. Getting dosage right is difficult (not for everyone, but it was for me). No thyroid means you are fully dependent on medication for the rest of your life, so it should be a decision of last resort.
That being said, I had chronic undiagnosed Hashimoto's, and my thyroid was removed in two surgeries (one for suspected but not confirmed thyroid cancer on the right lobe, and then another after the thyroid cancer was diagnosed). My thyroid was surrounded by swollen (noncancerous) lymph nodes because my body was very busy destroying my thyroid. My surgeon described a large amount of inflammation. At least 14 swollen lymph nodes were also removed from my neck during the surgeries. After the first surgery, I had a tight, uncomfortable feeling in my neck. After the second, I could immediately tell the inflammation was gone. Like the day after, I realized it felt different. So with Hashimoto's, it might be more than just the thyroid but the whole region surrounding the thyroid experiencing inflammation. If you can get Hashimoto's under better control (is that a thing? mine was diagnosed too late and my thyroid was a mess), it might help your neck feel better.
So... this is a permanent decision that will affect you on a daily basis for the rest of your life. I think all other options should be exhausted before choosing surgery, especially if it is not cancer. Getting to the right dosage of thyroid hormone if you are not already there might help reduce the swelling and allow less discomfort. Or, it might not. Sometimes thyroid inflammation is temporary, sometimes permanent.
Maybe the first step would be to talk to your doctor about the discomfort you are feeling, and make sure that nothing is interfering with your breathing or eating.
Good luck, and I'm so sorry you have to deal with this.