This seems like something you have to address to the priest directly, as it sounds like the secretary is only following orders from him. Maybe suggest using a coded register, so that attendance is taken down using code words for people, and the key to the code could be kept in a secret location.
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I very much doubt the priest wants to be doing this, but if the state came asking for the logbook and he said there wasn't one, I bet something very bad would happen to him and the parish as a whole, and if this is SSPX possibly the SSPX as a whole. We live in scary times.
