Is supplicating for complete healing from a chronic genetic disease, enzyme deficiency, considered an transgression in supplication? And is it permissible to pray at home when the intense heat of the sun causes pain to the patient, despite the mosque being close by?

There is no harm in asking Allah for healing from illness, and this is not considered an transgression in supplication (Du'a). As for someone who is harmed by going to the mosque during hot weather or if it causes him illness, this is considered an excuse that permits him to pray at home. It is also commendable for him to pray with his family in congregation. The reference point for harm is what causes hardship, similar to the hardship of walking in the rain.