I actually looked around a bit and stumbled onto a study that was pretty close to what you are looking. (COMPLETE luck....)
Anyway, a 220 pound person (100kg), for example was given 3 injections per week of approximately 7 IU (MWF). They tested thyroid function and didn't note a significant decline over a 12-month period.
Here's the abstract - I double checked my conversions and they seem correct.
Recombinant human growth hormone treatment at low doses does not significantly change thyroid function in growth hormone deficient adults
I actually wondered that myself when I took GH. They didn't study if the frequency of injection was a factor, but it's possible. Obviously, GH clears quickly, but IGF and other factors take longer. But two days between may allow the thyroid to remain "online."
I would like to find a study that did the same thing but for daily injections... I did daily injections myself and felt as though my thyroid was slowing - but it was just anecdotal (no testing at all). I felt I was becoming a bit sluggish and was actually feeling cold in rooms with people in which I had previously felt warm. I took t4 toward the end (last month of 9).
It feels as though my thyroid has fully recovered and I haven't taken GH for over 6 months.