Have there been efforts to show (using the Switching Lemma), for example, that SAT or 3SAT cannot have an AC$^0$ reduction to 2SAT? What are the issues or difficulties involved?

SAT and 3SAT are complete for **NP**, and 2SAT is complete for **NL** under AC$^0$ reductions. Hence such a proof would separate **NP** from **NL**.

In the case of a 3SAT to 2SAT reduction in AC$^0$, the bottom fan-in is bounded by 3 (at least prior to the first switching).