Negotiating salary before in-person interview for an administrative position

Not sure what your question is. You could respond yes I am interested in continuing the process and see where it leads. Maybe they have some benefits you don't know about. Maybe the job is even cooler than you think. In other words maybe something happens and you are willing to take the job for less money. Then again, maybe things pan out how you expect and you tell them know that you have seen the complete offer and understand the job more, you cannot take it for less than X (whatever that number is, plus some wiggle room).

You could also respond no, because do you really want to put up with crap like that.

I would just take their statement at face value. In my (admittedly somewhat limited) experience with academic hiring, stating a salary range is done simply for the obvious reason: so that if the candidate is unwilling to work for that salary, they can drop out immediately, thus saving the university (and themselves) the time and expense of further interviews.

If you are willing to consider working for a salary in that range, then go to the interview. If not, then withdraw and look elsewhere.

As far as "take the job and hope for a raise", note that academic institutions are not generally known for being able to offer large raises to reward or retain valuable employees. I wouldn't count on that.

It's obviously not true that nonprofits are inherently unable to negotiate on salary, and I don't think that's what your HR contact meant. I just take it as "we're a nonprofit and our budgets are tighter than they might be in industry, so our salaries are therefore lower".