It seems to me that this particular person wants to look at the books to get something on the pastor; thus her request for his personnel file. The article also doesn't say that she donates any money herself.
The fact that a conservative bishop, DiLorenzo, backs this pastor with his diocesan attorney indicates to me that the lady has too much time on her hands, and the bishop would rather she spend it somewhere else.
That does not explain why she should not be able to look at the financial reports. You should be careful to not ascribe pure motives to the Bishop. Not saying he isn't honorable, but even otherwise honorable people do things out of character for reasons they'd rather not share.
In the end, it is my guess that she has good reasons to be suspicious.