The thermostat is 24V, not 12V. All low voltage control components in a HVAC system run on 24V.
I think you should connect your thermostat adapter wire as follows:
You want to make sure all power is off to both the inside and outside units before unscrewing any wire nuts. Have extra wire nuts of the proper size handy, and wire strippers, in case a wire breaks off in the wire nut when you unscrew it. Also, make sure you have a flat screwdriver small enough to fit into the terminal strip on the wyze adapter.
Remove the wire nut from the gray and white wire and insert the C wire from the adapter into the connection and reattach the wire nut (Note: this is the only case where you are only adding a wire, not removing any from the wire nut).
Remove the green thermostat wire from the green wire wire nut, and attach the G lead from the adapter to the green lead from the fan switch with a wire nut. Attach the green thermostat wire to the G terminal on the adapter and tighten the screw.
Remove the red thermostat wire from the purple and red wire nut, and insert the R wire from the adapter into that wire nut with the purple wire and tighten. Insert the red thermostat wire into the R terminal on the adapter and tighten the screw.
Remove the white thermostat wire from the white heating element wire nut, and attach the W lead from the adapter to the white lead from the heating element using the wire nut. Insert the white thermostat wire into the W terminal on the adapter and tighten the screw.
Remove the yellow thermostat wire from the red and yellow wire nut, and insert the Y wire from the adapter into that wire nut with the red wire and tighten. Insert the yellow thermostat wire into the Y terminal on the adapter and tighten the screw.
This should allow your Wyze thermostat to get a solid 24V connection even without a dedicated common wire in the thermostat wiring.
Ideally, instead of the rats nest of wires, the installer would have used a terminal strip to get all the wires in one place, and then used zip ties to clean up the wiring loom so that it wasn’t so hard to see what was doing what, but unfortunately some people are too rushed to do a good job.