I am not quite sure that sugatagarbha/tathagatagarbha quite means a priori perfection. I believe those terms refer to potential for perfection.
Buddha-nature (buddhadhatu) might be the term AL is looking for.
Depends on which school of thought you subscribe to. Some consider it ( a priori) pure and perfect, others consider it as a potential. This is the first time I have seen dhatu translated as "nature", normally it means sphere, matrix or realm.
You're probably right that it depends on the tradition. I have seen tathagatagarbha and buddha nature used interchangeably, or even the latter as a translation of the former.
Regarding the "nature" translation of "dhatu", it is a peculiar translation. I wonder if "nature" is a twice removed translation of "dhatu" ie. Sanskrit -> Chinese -> English.
Dhatu does seem to fit the Tientai conception of Buddhanature - a rather involved explanation involving what Zhiyi called "mutual possession" of Dharma Realms. Anyway, that's why I premised my response with the lens through which I was approaching the question. Many terms various traditions use in common can have very different meaning.