Definition of clinical: relating to the observation and treatment of actual patients rather than theoretical or laboratory studies. Before the TSH test was used in 1973, doctors treated patients based on clinical symptoms. While the TSH has value, this test has flaws from the reference range itself to conditions that prevent TSH from rising in response to falling levels of thyroid hormone in the blood.
In Canada I expect that it won't be easy but you are going to have to feed your doctor enough information that conflicts with his training and inherent beliefs (not to mention the pressure from the NHS), to get him to read and re-consider what he has been doing for years, no doubt.
Starting right from the definition of hypothyroidism, it is not just inadequate thyroid hormone. Instead the correct definition is "insufficient T3 effect in tissue throughout the body due to inadequate supply of, or response to, thyroid hormone." So diagnosis for potential hypothyroidism should not be based solely on biochemical tests. There are no biochemical test that can be used as pass/fail decisions about a person's thyroid status. The person's thyroid status is dependent on the level of Tissue T3 Effect. So the best approach to diagnosis is a full medical history, followed by an evaluation for signs/symptoms that occur more frequently with hypothyroidism than otherwise, and then extended biochemical tests. The tests that are necessary are Free T4, Free T3, TSH, Reverse T3 at least at the beginning, cortisol, Vitamin D, B12 and ferritin.
The most important diagnostic is the signs/symptoms I mentioned. TSH is useful as a diagnostic only when at extreme values. Intermediate values for TSH have only a weak correlate with the biologically active thyroid hormones, Free T4 and Free T3, and a negligible correlation with Tissue T3 Effect which represents the patient's thyroid status. So how is it supposed to magically represent a person's thyroid status? Clearly it does not.
In using Free T4 and Free T3 for assessment, you have to take into consideration that the reference ranges are so flawed, and skewed to the low end, due to the erroneous assumptions used to establish them. FT4 and FT3 have only a weak correlation with Tissue T3 Effect.
Reverse T3 in excess can adversely affect Free T3. High or low cortisol can also affect Tissue T3 Effect. Also, Vitamin D, B12 and ferritin fall into the category of affecting response t thyroid hormone.
There is more to discuss, but I have to leave for a while. When I return I will provide more info. In the interim I suggest that you click on my name and then scroll down to my Journal and read at least the one page overview of a full paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.