Could it be that the bore snakes need to be really wet to work good and not get stuck in the barrel. Maybe a good dose of Hoppe's or other bore cleaner from the front to the end of the brissels and a good dose of clp or other treatment from the brissels to the end of the snake ? That way the snake is well lubricated ?
Yes, the snake should not be pulled through the barrel unlubricated unless absolutely necessary . It doesn't need to be soaked, and infact probably shouldn't be. I use a bit of bore cleaner in front and behind the bristles and depending on how much oil I want in the barrel I'll put extra cleaner at the end of the snake to coat the barrel for storage. If im planning on firing the gun I use much less oil. And I use the snake until it is just a very VERY thin layer of oil in the bore.