Actually, you may get an increase in FT4. How do you KNOW it will lower FT4? Your GH use may actually "cure" your central hypothyroidism, or aggravate it. You don't know until you run consistent labs. And likely, over several months, you'll see changes. Possible, they can decrease one month, and increase the next.
Assuming you have a "baseline," because obviously, it sounds like you've tested and know you're hypo. Don't rush to introduce exogenous thyroid. To put it bluntly, you want to try and avoid tinkering with your thyroid at all costs... it's a last resort to do so. Give the GH time to make "alterations" and settle-in.
As far as muscle pain, how is GH deficiency causing that? Are you referring to recovery (as you eluded to in your post)? GH deficiency just doesn't necessary cause muscle pain. Fatigue, joint pain, I guess muscle pain is possible.
Maybe I should've asked this in the beginning... how do you know you're GH deficient?