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Egg laying trait is a Polygenic trait. So males transfer their sex linked genes(likely 2) to their doughters. Now I calculate that the total genes involved is about 20 since the reciprocal cross difference is only 10%

That's a good point, the F1 back to the native chicken would increase egg production on native chickens. You can take the same F1 rooster and cross it to both Native and Production type egg layers, the Back Cross to any of those will surely produce better egg layers than the natives.

Line breeding would mean you are using the F1 male back to it's mother which is not always recomended specially when dealing with prolific egg layers, in this case Back Crossing to the same breed females achieves the same egg laying performance without any of the genetic issues that may appear...

This is a very straight forward breeding project, Back crossing the Naked necks back to the desired parent will improve egg production.