Good for your GP - that's rare in itself but I cant understand why she didn't start you on treatment herself. She is obviously not willing or able to start you on treatment whilst you TSH is still, only just, in range.
With a strong family history, high antibodies and high TSH I don't know what more evidence your endo would want to start treatment. If I remember rightly the earlier you have your TSH test done in the day the higher the reading will be so maybe remember that for next time - always book an early appt for bloods and your TSH will be higher - which will help your cause. If I've got that wrong I'll come back to you or someone else may correct me. Can you remember what time you had your bloods done?
I can only stress again that you may need to fight your corner and maybe, IF he/she refuses treatment, ask how ill you will have to become before he/she will consider it.
Ask the endo to test your B12, vitD, iron and ferritin for you as these can often be low. That may be easier said than done as different endo's again have different views on testing. My endo has thrown ever test under the sun at me but others on here have reported that their endo's refuse, but if you don't ask you don't get - mention that you have heard that people with this illness/high antibodies can be low in the above and that is why you are asking for them.
You may be pleasantly surprised and the endo might know the obvious - if he doesn't treat you now you'll only be back with 6 or 12 months - and start you on treatment straight away.
Do as much research on the condition as you can so that you can hold your own with him and you'll be fine.
Good luck and let us know how it went.