A T4 'within range' does not rule out the possibility of low thryoid. This test measures the amount of the hormone in the blood, not the amount that is actually being used.
One of my IGs in a pet home was [to me] obviously low thryoid, but his T4 was always within normal range. Further testing did determine the problem and, on correct dosage of thyroxine, his symptoms have gone
See his story at
http://merikaez.4t.com/Nutmeg.html
From what you say, low thyroid does seem very likely in your dog's case. Have the thyroid tests done. I see no reason why your vet friend should object... its quite normal and acceptable practice to get a second opinion. If she can't cope with that, its her problem, not yours.