Your message made me think of four things that might help.
First, since your GD started after your pregnancy I thought it might be related to that. The following website on thyrotoxicosis has pregnancy listed as a cause. If it is, maybe your treatment would be different?https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1476727/
Second, I'm also being treated under the VA system. Did you know that when you ask for an appointment you can request a different doctor? I had an endocrinologist that was young and arrogant. After struggling with him for a year I finally asked for a different doctor. When asked for a reason I stated that we had a “personality conflict”. It was true and they could not argue with it. They gave me an older woman who was very knowledgeable and much more sympathetic.
Third, have you had an iodine scan? I'm not sure what it is called, but they give you iodine (I had an injection) and then they scan you with an MRI or PET scan. The iodine goes into your thyroid gland so they can see what is going on. When you talk about your uneven symptoms it makes me wonder. Maybe this test would help.
Fourth, I've found that the Messaging system in the MyHealthyVet website is very helpful. When I had some difference in my status I messaged my doctor and asked for a blood test and a possible change in my dose. She authorized the blood test and after a couple messages back and forth, lowered my dose. I think they must monitor the messages somehow. They have a warning not to have urgent stuff, but they seem to answer promptly and courteously.
Finally, it made me sad when you said you feel hopeless. The military is like a big family! Talk to one of the other moms on your post or even talk to a woman when you shop at the commissary. They probably have been through something similar or know someone who has, that you can talk to. Having your husband on deployment when you have a baby can be really hard. You aren't alone, reach out, there's other moms out there who would care and can help you.
I hope that helps and that you feel better.