woah, 6 is too high. I depends on the reference range for your lab, but that sounds well over normal to me. Do you know your T4 levels?
Is this an endocrynologist or GP? It is very tricky if you're going hyper and hypo, though. Definitely see a specialist if you haven't already - sometimes there are treatment options. Thyroxine is fine while breastfeeding, but you doc's concern may be if you switch to overdrive again. Treatment for Hyperthyroidism I think might not be ok when breastfeeding.