Even though tests show that you have Hashimoto's Thyroiditis, and have hypothyroid symptoms, your doctor appears to be an advocate of the "Immaculate TSH Belief". By that doctors only pay attention to TSH, in the mistaken belief that it accurately reflects thyroid status. That is very wrong. TSH is a pituitary hormone that is affected by so many things that at best it is only an indicator, not a diagnostic, to be considered along with more important indicators such as symptoms and also levels of the biologically active thyroid hormones, Free T4 and Free T3. Note the following quote from an excellent thyroid doctor.
"The free T3 is not as helpful in untreated persons as the free T4 because in the light of a rather low FT4 the body will convert more T4 to T3 to maintain thyroid effect as well as is possible. So the person with a rather low FT4 and high-in-range FT3 may still be hypothyroid. However, if the FT4 is below 1.3 and the FT3 is also rather low, say below 3.4 (range 2 to 4.4 at LabCorp) then its likely that hypothyroidism is the cause of a person's symptoms."
A good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resulting TSH levels. Symptom relief should be all important, not just test results and especially not TSH results. You can get some confirmation of this by reading at least the first two pages of the following link. If you want to delve further, then read the discussion and scientific evidence supporting all the recommendations. If your doctor is reluctant to treat clinically, then you can also give a copy to him and ask him to read and then consider clinical treatment as described in the paper. Typically optimal levels are Free T4 at least at mid-range, and Free T3 in the upper part of its range, adjusted as needed to relieve symptoms.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Also, hypo patients are frequently deficient in Vitamin D, B12 and ferritin. all three are important, so I suggest testing and then supplementing as needed to optimize. D should be at least 50, B12 in the upper end of the range, and ferritin at least 70.
Tsh 3.12 (.40-4.50 miu/L)
T4,Free 1.0 (0.8-1.8 ng/dL)
T3,Free 2.8 (2.3-4.2 pg/mL)