Could a descriptive complexity version of Rice's theorem be used to separate AC0 and PSPACE?
Deciding the nontrivial properties of (an indexing) sets in a complexity class is as difficult as computing the graph of the universal function for the class. Intuitively this means that the only way to decide a nontrivial property is to simulate the machines and wait for answers. It seems to me that the idea of using such a property will just give what is known by the hierarchy theorems. (See theorem 4.2 of D. Kozen, "Indexing of subrecursive classes", 1978 for details and the exact statement of the theorem.)

We can decide the $grU_{AC^0}$ (the graph of universal function for $AC^0$) in $PSpace$, the reason is simply that $AC^0 \subseteq L$ and we have a universal machine for languages in $L$ in $PSpace$, so it is easy to simulate the $AC^0$ machines (or the descriptive complexity equivalent of $AC^0$ which is $FO$ queries) in $PSpace$. This means that that we can decide the property you have stated in $PSpace$. Since it is a nontrivial property, it is not decidable in $AC^0$. So this argument separates $AC^0$ from $PSapce$. 

But is this surprising? No, since we already know a simpler way of stating (essentially) the same argument: $AC^0 \subseteq L \subset PSpace$, the last proper inclusion is by the space hierarchy theorem.