Maybe start with a local Jewish high school - if OP is in Cleveland, there are non-yeshiva/non-beis-yaakov Jewish high schools which certainly have non-shabbat and non-holiday PSATs this fall. Unclear if they’d allow her to take it there as you noted.
BTW the Jewish holidays are weekdays this year and with the October overlap, this could come up at nonJewish schools.
You are right that some schools are restricting seats. My son needed to take two AP tests that were not offered at his private (Jewish, non-Orthodox) school. One of them he could do where I worked, but that was a special case. The other also wasn’t offered where I worked, so we looked around. Another Jewish (Orthodox, not yeshiva) school said OK but ultimately so did our hometown public school so he went there instead since it was closer. My point is it may take some research.