How to create a smart 3-way positional switch?
A 3-position switch would be implemented in z-wave by independent 2 on/off switches, and the &quot;3-position&quot; function is programmed as a &quot;radio-button&quot; activating one or none (but not both) of the 2 on/off switches.
If it's ok to switch one or the other mode, and no hazard exists if both are on, as in your case (no harm if timer is on and switch is also on), there are two-relay modules that could fit inside the electrical box of your current switch.
Have a look at this:
https://www.homecontrols.com/Qubino-Flush-Relay-QUZMNHBD3
If you do not want to use a smart-hub, then you could try it with a &quot;remote switch&quot;. There are models that provide one remote to control several outlets, e.g. for outdoor lighting.
https://www.amazon.com/Etekcity-Weatherproof-Electrical-Unlimited-Connections/dp/B078YL37MM?ref_=fsclp_pl_dp_12
If you are somewhat handy (and I think you are, if you are thinking about the Arduino), you can wire the double-relay unit so that it overrides or replaces your wall switch.
Since you are using an independent timer as one connection option, I don't think you can use these remote receivers straight out of the box. You have to open up the case and modify the wiring at the switching poles of the relay. Basically you reverse wire the switch: the relays have to select between two input plugs (one powered permanently, one powered through the timer), and supply one socket.
Simply reversing the input plug and output socket of the entire unit would not work because the unit derives its own power from the plug side.
I can update this answer with a sketch if you need one.