I have Hashimoto's, and I've been on thyroid replacement for at least 15 years. All my symptoms went away within about 2 weeks of taking thyroid.
I didn't have the same symptoms as you, mostly extreme exhaustion and a few other minor things. (Prolonged, irregular period, anxiety, can't remember what else. Oh yeah, and cold all the time, as mentioned below.)
If the Hashimoto's is causing your symptoms, it makes zero sense that they are "not bad enough to treat." You need a clearer answer from your doctor.
Tell him, if Hashimoto's is causing this, then I want to go on thyroid replacement, because this is very unpleasant and I don't want to live like this. Once you start taking thyroid, you will know very soon whether Hashi's is what's causing your symptoms, or not.
If you take thyoid and those symptoms don't go away, then it's something else, and the doctor needs to get to the bottom of it.
The doctor absolutely should not dismiss this and refuse to treat you. There is no reason why a doctor shouldn't give you a thyroid prescription. It is not a big deal to take Synthroid or Levothroid. It's a pill, once a day, no big deal. You need to insist.
If your doctor doesn't help you, go to another doctor who will.
Aches and pains and sore joints can be a lot of things. Try not to Google it, it will scare the bejeesus out of you.