Option 6 would be scrap it. Attempting engine swaps on Honda self-propelled mowers can be a pain. In order for it to be easy, finding a GCV190 would be best, and also difficult to find. You would be better off buying a used Honda mower over most of the options.
You are going from a Kohler K series engine (50 years old) to a Briggs twin engine. Wiring won’t be your only concern. This will require modifications and would look hard at it before attempting.
Good pre-mix oil such as Stihl synthetic (gray bottle) is more like $5 bucks per 2.5 gallon bottle, not $2 dollars.
Ethanol is hygroscipc (meaning it attracts water). All I run is 10% ethanol in all my equipment. Small engines will not tolerate 15% ethanol. Most people think ethanol is evil and...
Look like a Honda GCV160 engine with no push rods. I am sure this engine costs more to build, and therefore main reason it never caught on. I have never seen a Briggs DOV in person. Watched most of the video, interesting.
With a new aftermarket carburetor, and the engine is hunting and surging, it is running lean (not enough fuel). The air/fuel mixture ratio is not right. Best thing you can do in this case is completely clean the original OEM carburetor and reinstall.