tsh suppression occurs when t4 levels are high but in her case its still low.
It is not a pituitary issue. Suppressed TSH is the frequent result from significant daily doses of thyroid med. Our bodies evolved expecting a continual small flow of thyroid hormone from the gland, not a large dose once a day. You are correct that TSH should basically be ignored when taking thyroid med.
please tell if its actually free t4 or t4, similarly for free t3, t3 etc also provide reference range.
for the time being i suppose the results you gave is like this
tsh .03 which is too low
free t3 (not t3) 1.24 which is also way too low
free t4 (not t4) which is borderline low
by seeing this result it seem she have pituitary issue and require to up the dosage of levothyroxine and need to completely ignore the tsh value as its giving faulty signal to thyroid gland thus also not useful for prescribing levothyroxine dosage.
Is that T4 test for Total T4 or Free T4. They are quite different. Also please post the reference ranges shown on the lab report for those results. Most important, what symptoms does she have, if any?