How can I prime for bottle conditioning without oxidizing the beer?

Oxidation due to the introduction of oxygen during packaging is an important concern for all packaging methods EXCEPT for when you are priming and bottle conditioning.

Yeast cells that are actively consuming sugars in the presence of oxygen actively and efficiently consume all of the available oxygen. Since you are priming (adding sugar), you can count on your yeast to consume the oxygen while they produce your carbonation. The same argument holds if you are priming and conditioning in a keg (assuming that you serve from the keg using pressurized carbon dioxide to push the beer out of the keg, rather than a hand pump).

If you are force-carb (or even prime and condition) in a keg and then fill bottles with a counter-pressure filler, you absolutely should be worried about the introduction of oxygen.

Given that you can count on any (reasonable) amount of oxygen that is introduced while priming to be quickly eaten by the yeast, you should make sure that your priming sugar is well mixed into your beer before filling your bottles. Lack of proper mixing is a much bigger risk to your final product than over-mixing (and introducing extra oxygen).

It's actually going to be pretty difficult to cause excess oxidation if you aren't doing it intentionally or performing extra transfers. If you've had uneven priming then you are going to need to up your mixing game. Jeff's method is going to be foolproof for all methods that aren't just adding syrup to your mix.

Maybe some context will help ease your mind/paranoia. I would argue that your transfer from carboy to bottling container is the largest oxidation risk as that leaves the most surface area exposed which is why we leave the tube in the liquid during transfer when possible and not running down the side. The surface area to volume ratio in a 1/16th inch deep stream running down the side of a bucket is really high. In a bucket, only the very top layer is exposed to air even with gentle stirring. It's almost nothing compared to the total volume.

It's like a fish tank. Without intentional aeration of some sort fish will die due to lack of oxygen in a relatively short period of time. Vigorous splashing or mechanical bubblers are the only way to put enough oxygen in solution for survival. While not exactly the same it gives an idea of the lengths needed to meaningfully increase oxygen levels.

I always boil and cool my priming sugar solution and add half of it to the bottling bucket. I rack about half of the beer into the bucket and then add the rest of the priming solution as the remainder of the beer is flowing.

Afterwards, I use a [sanitized] brew spoon to very gently stir the beer in the bottling bucket. As long as you avoid churning it and don't introduce bubbles, you'll be fine. Yes, there's always a slight risk of infection, but as long as the spoon is properly sanitized you're not risking a whole lot, and I'd much rather make sure the priming sugar is evenly distributed than risk bottle bombs later on.

The two methods I've used have been:

• Put the sugar solution in the bottling bucket, then rack carefully to that from the fermenter. The movement of the incoming beer is plenty to mix in the sugar.
• Siphon or carefully pour the sugar solution on top of the beer, then stir it with a sanitized spoon very slowly for ~5 minutes.