Classic pull out method, it'll get it done but not the most effective. Works most the time but not the safest...
He means put the cleaning rod in through the muzzle, thread on your brush then pull back out... Works fine for brushes but won't work for a jag/patch and you can't use a bore guide. It also pushed debris into the throat and chamber which I don't like.
I ran into the same problem and I was never a fan of the bore snake dragging dirt through the barrel. Is your cheek piece removable/adjustable? Mine was, so I just marked the height with a piece of chalk pencil that I keep in my cleaning kit, remove the cheek, clean and reinstall. Not the most convenient option, but if you can lower the cheek piece, I found that a little chalk mark got me back to a great sight picture every time.
my carbon tipton bends enough and is long enough that i can use it around my cheek piece and do not have issues. i also use Lucas custom bore guides for each rifle and have no issues.
I am a precision rifle competitor and I can tell you that the Tipton rods and Lucas bore guides are simply the best.
My comment to the OP would be to alter or replace your stock. You are simply asking for damage to your barrel and action by cleaning from the muzzle end or using the Otis cleaning products. The crown of the barrel is one of the most important parts of the rifle in regards to accuracy. One should ALWAYS clean from the chamber end and push all of the cleaning materials towards the muzzle. And trying to bend the rod into the chamber is causing unnecessary wear on the throat.