I can't imagine AA shipping a wood stock that has not been sealed or finished with some material. Many times stock finishes are referred to as "oiled" or "hand rubbed", which only means the stock has an appearance similar to an old-fashioned hand-rubbed finish produced with a material like linseed oil. An "oiled" finish is not glossy, and may or may not leave the pores visible or unfilled depending on the thickness of the applied finish. My recently received Anschutz 1761 has an "oiled" finish that is matte in appearance, but perfectly smooth, with no pores visible. It is my understanding that the AA walnut stocks have a finish that is so thin that most or all of the pores are still visible on the surface of the wood.
Factory gun stocks used to be finished with conventional lacquers. These would dry quickly, but were not very durable and were succeptible to softening from skin oils. I do not believe these are used any more due to them containing high percentages of VOCs. Not too long ago I had a small table top refinished and was told the finish would consist of two coats of a gloss catalytic finish, followed by a top coat of the same material in matte. I think it is likely that a very thin application of a similar modern furniture grade finish has been applied to your stock. If you do not want to change the appearance of your stock, you only need to wipe it down after each use to prevent the accumulation of dirt and skin oil. I use the Birchwood Casey silicone impregnated cloths sold for use on guns and reels. Modern finishes are very durable and do not generally need protection. My only concern would be that the applied finish is so thin that it is not water resistant, although I don't think that would be the case. The silicone in the Birchwood Casey cloths will add another layer of water resistance if that is a concern. I have used them for decades on stocks with both lacquered and more modern finishes with no problems whatsoever.
I am more confident in the compatibility of silicone protectants with wood finishes than I am mineral oil, if that is the major component of Ballistol. To remove any remaining mineral oil, you can just wipe down the stock with a rag lightly wet with mineral spirits.
I use Minwax Antique Oil on unfinished wood, and to restore finishes on furniture and gun stocks. It is a linseed oil based product with driers and hardeners added, similar to TruOil. Applied over an existing finish it will build gloss quickly since it will not soak into the wood. I would not use it if you want to retain the current appearance of your stock.