Hey
This is tricky as you say. The dr is following the NICE guidelines and most hospital management algorithms for possible subclinical hyperthyroidism. Here's a summary of the NICE guidelines:
Basically, you have to have a goitre/nodule or symptoms of high thyroid hormones (which you have) + 2 low TSH tests done at least 3 months apart. Without that, you will struggle to be referred to an endocrinologist. Insist on that second test in 3 months and if the level is just as low, and your symptoms are still there, then you should be able to justify a referral to an endocrinologist. That said, if your thyroid levels aren’t explaining the thyroid nodule/goitre, then you should be referred for further investigation (ultrasound most likely) to rule out anything sinister. Your SHBG is too close to the reference range to be clinically relevant I would imagine. The tests are a bit unreliable if you are taking medications for an acute condition or in hospital with illness, etc. If any of that is remotely applicable, there is certainly reason to test again or at least refer to an endocrinologist.
I know it's not what you wanted, but hopefully the lump is still being actively investigated? I would imagine when that is clearer, they will refer you to an endocrinologist anyway. At the very least, you have a good argument for another TSH/T4/T3 test in 3 months.
Wish I could've been more helpful.