High LH and low T typically means that the testes are a problem and such problems can only be surgically fixed if there is a correctable vascular problem. Doctor can check for that. Any ache in your testes.
Another problem that cannot be surgically corrected is testicular cancer and that is a young mans disease. There can be FSH secreting testicular cancers your low FSH rules that out.
Typically, FSH and LH levels are similar, so your high LH and low FSH seems quite odd, so that needs consideration.
When LH is high and T is low, hCG which acts like LH will not work any better.
I would retest LH/FSH to see if this pattern still holds before taking any other action. LH is pulsatile, so it can change by the hour. FSH is typically the better indicator of LH status than LH itself, but you may be an exception to that.
LH secreting pituitary adinomas are rare, but still happen.
HCG secreting tumors can easily be detected in males as background levels of hCG are typically undetectable.
If LH levels are high and constant, that can turn the LH receptor pathways off.
The above is a brain dump may be useful to know if you get into any complicated diagnostics.
T4 is well below mid-range.
uptake and indexes are basically obsolete now that fT3 and fT4 labs are available.
TSH looks great, body temps are low.
As fT3 is the active thyroid hormone that is part of the body’s temperature regulation control loop and T4 is low, I expect that fT3 is below mid-range. Note that T4 is a reservoir for fT4–>fT3 generation.
Head trauma can damage the pituitary, but the result would be expected to be lower LH and FSH, not high LH, low FSH and low TT.
Taking higher amounts of iodine increases TSH. More to the point, please observe changes to mood, energy and body temperature. You are mostly on your own with these things as everything is “normal”.
When iodine has been low for a long time in the past, high TSH can lead to thyroid nodules that start making thyroid hormones independently of TSH control, thyroid levels improve and TSH starts to go down and as things progress to hyper, TSH will pass through the 1.0 zone, looking perfect, later on TSH gets very low. So there can be complexities that most doctors will not consider until things get abnormal. Again, this is a brain dump that might not have any bearing on your case.
You have not had FT tested and that may be abnormal. Diagnostics need to consider symptoms and other things. As SHBG was not high, your TT is probably not inflated by a disproportionate amount of T+SHBG.
If you want to nudge your T levels down, try taking OTC Tagamet /Cimetidine. It can interfere with E2 clearance, increasing E2, which would normally decrease LH and T to some extent. But with your high LH and low FSH with low T, all bets are off.
You stated that after talking to nurse, that fT3, fT4 would be tested. What happened?