New cases always* have dings in the mouths, Lapua probably the likely exception. I would not use a perfectly round sharp champfer tool to cut material from an out-of-round case mouth. You're going to remove brass inconsistently, depending on the size and location of the dents. I'd want the case mouth to be round first before I started removing material. Why? Because CONSISTENCY matters. It won't blow up your gun or anything like that, but if you are going to do it, why not do it CORRECTLY?
There's a reason all manuals follow the same basic step by step procedure. There's a reason the steps are in a particular order.
I also would not "partially" size the necks, because CONSISTENCY matters, how does that effect neck tension, and how inconsistent is your batch if you only partially sized SOME of the case mouths but not others? No thanks. I want every single piece of brass to be clones of each other, as close to identical as I can get them. That means sending them all through the same exact prep process every single time. And it's not that hard to just do it right vs taking shortcuts.