I find the JW statement to be credible. (This is about the only thing about the JW’s that I find credible). Completely free access to the UN library appears to be limited to online resources. It sounds like physical, in-person access is somewhat more limited. Not sure whether NGOs qualify as part of the permanent mission staff, but if so they have pretty much automatic access. Online access would not have been available in 1991 when they registered as an NGO, and the library would be a handy resource to the JW organization, which is just across the river in Brooklyn. Finally, it seems likely that access rules have changed in the last 15 years, perhaps even in the last 5 since the JW letter that you posted - NGO membership may have more explicitly allowed easy access to the library.
So I accept the explanation of the JWs of their reason for registering as an NGO, and their reason for withdrawing that registration. Having said that, it seems like a poor move PR-wise. And it may well be hypocritical, although I don’t know enough about their stance on the UN to judge that fairly.