If the carburetor is a clone of the carbs used on the honda GX series engines it might have a removeable pilot jet which looks like this and going a tad larger might smooth it out.
The only way to correct the low throttle surging is to richen up the low speed circuit which is why running a little choke smooths it out.
You might be able to go larger on the main jet which might add just enough fuel on the low end to smooth it out as well. Typically all these engines error on the side of being too lean for emissions so it might even gain a little power with a larger main jet.
Then I bet it would probably smooth out with an air filter. It would pose just enough restriction that it would richen up a touch.
Obviously that's not a feasible solution, so enlarging the pilot a tiny bit would do the trick. Measure it, figure the area of that circle, then add about 5 to 7 percent and it'll run smooth as can be.