Wiring for Split Outlet with Z-Wave Switch?

You can also consider using something like this:

https://www.amazon.com/gp/product/B08R79686G/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&th=1

SONOFF ZBMINI ZigBee Mini Smart Light Switch(2 Way), Compatible with Alexa, SmartThings Hub, Google Home & SONOFF ZBBridge, ZigBee 3.0 Hub Required, Neutral Wire Required, 4-Pack

From your description the switch is just breaking the hot side of power to the outlet: Power comes into the outlet box, the hot will be connected to the black wire going to the switch, at the switch that black wire will be connected to one connection and the white wire will be connected to the other connection on the switch, then back at the outlet, that white wire will be connected to the hot connection on the outlet. That white wire is supposed to be marked with a ring of black tape or some other mark to indicate it’s not a neutral but that doesn’t last very long typically.

With this relay, you, in the outlet box, disconnect the hot from the switch wire and then connect the two wires from the switch to the switch connections on the relay. The hot and neutral are connected to the load in and neutral respectively, and the load out and neutral are connected to the outlet.