An item in the CSC shop is listed as part of the sale (Delish Heat Bottled Sun), but when I go to my cart, the sale applied to everything EXCEPT for that item. Either the sale is falsely advertised for this item or the discount just isn't applying to that item specifically.

That definitely looks like a glitch

This is a bit late, but have you tried removing the item from your cart and re-adding it? I've seen some other people having this issue too, but redoing that seemed to resolve it.

That worked! Thank you!

Yay, glad it worked out for you! :D