I think I get it now, and I believe it may still be doable.
May I ask, is this page: http://tst.lasalle.edu/programs/arts-and-sciences/
just a regular page, or some sort of category archive or something else? I'm asking, because there is an extensive plugin API, which allows executing code during the search process, and modifying the query arguments. If somehow there was a way to get the desired category based on that page ID, then it should be very likely easily solveable.
There is actually an option on the back-end under the Advanced Options -> Exclude results panel, that allows including results only from specific categories. In this case however it would only work, if the category filter on the front-end did not have the 'Choose one option' selected as default – because that automatically means that inclusions should be ignored.
I'm thinking, if this option had the first set of categories choosen (or the only one you want), and the actual front-end filter only had the second set, without the 'Choose one' option. Then the plugin would automatically restrict to the first set defined on the back-end, and to the one that is selected from the drop-down as well.
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