Since I started this thread, I feel like I should add my conclusions. After putting in over 100 flights on my Nuance with a 2 bladed prop, I accept what I learned about 40 years ago and that is, when you pull (pitch) you need a little right rudder and when you push you need a little left rudder. I tried everything to trim that phenomenon out, even sealed the aileron gaps as Tony F suggested. I got so I was pretty automatic and was flying fairly well.
Humans are very adaptable. When I was about 12 years old I bent the fork on my bike and it always pulled to the left. I got used to it and when I rode someone else's bike I felt like his was pulling right. So you can do it. I remember Dave Wilson saying it is better to have a plane that always needs the same inputs than one that you need to watch to see which way it is going to veer.
The motor on the Nuance has been very noisy so I decided to take my Gaudius with its Contra out this past week. Wow, no rudder required during pushes and pulls. What a joy.
Conclusion: You can learn to fly a 2 bladed prop but it is a lot easier to perform the maneuver you picture in your mind with the Contra.
Jim O
It's interesting that you accept having to apply rudder during pulls or pushes as normal, as P factor mostly affects high angle of attack situations, which we don't have flying pattern. It is a trim issue causing this.
I have only used 2 blade props and do not have to apply rudder during pulls and pushes.