>> Heavily compacted, clay soil with new, lush sod installed on it about three-ish years ago.
That is a little bit like installing sod over concrete. It may live, but would have preferred to have had some soil.
I don't know enough to say what the fastest or most efficient thing to help is. If there is a bug or disease infestation, identifying and fixing that might be top priority. Sending a photo to your local extension service would be sure to get an expert answer.
BUT ...
Starting with clay, you won't have ANY soil until you add enough organic matter to feed enough soil organisms to make soil from clay. Top-dressing is probably the only way to add it (or thick compost tea). If it is just UN-composted leaves, at least shred them first and scatter thinly, but repeat. Mow them in. I don't know whether it is possible to sheet-compost over sod. I would guess "not possible, that kills the grass".
Maybe sod doesn't need soil, but if it does, your clay needs compost. Or any organic matter that will decompose and feed the microbes and worms and so on.
