The router will give you the best finish, But with a router, its like painting something, preparation is everything.
You must hog out almost all of the inside wood first. Forstner bits and even chisels are good for this.
Flush trim router bits work best when they are only removing an 1/8" or so (at the most, less than half the diameter of the bit).
If you try to plunge the flush trim bit straight down into walnut, you will have an instant snakes and ladders board (minus the ladders of course)
Make the template deep, so that you can take three or even four shallow passes to reach depth and still have the bearing resting on the template.
Oh, and one more tip if you dont have much router time in.... Direction of cut is everything. If you go around the template with the blade spinning backwards, then CARP is the only outcome on walnut as it will shatter.
Hand held router on an inside circle you should start cutting and then move to the right until you get back where you started from.
Its a faff, but the end result will be perfect and well worth the effort.