You have gotten some well-intentioned but incorrect advice here.
I have run into this before, repeatedly, not just theoretically, in terms of type of problem, but specific to Bergs (especially really good playing Bergs on 10Ms).
Here's what's going on: your mouthpiece has too little volume in the chamber and baffle for the horn, and for you.
For someone else, who plays "flatter" inherently (tighter embouchure, narrower body-cavity, whatever) the mouthpiece will be pushed in further. For you you're far out because you play sharper.
It is common for less accomplished players to play sharper, and for more advance players to play flatter, so that, in other words, more advanced players with more relaxed, more advanced technical tone production tend to push in further with the same mouthpiece. But body cavity and other things are also factors.
Regardless of what you do to the cork, you're going to tune in the same spot, unless you change how you produce sound.
You can stabilize the mouthpiece in that spot -- whatever spot it is that's good for you now -- by having the cork redone. But if it's very far out, and it looks quite far out, you won't be as steady physically as if the mouthpiece was a better match for you and for the stage your tone production is at now, which may change later (the mouthpiece will be less wobbly regardless of the cork when it's pushed further in, more wobbly further out).=
Remember: the problem is that you are sharp. That's why you have to pull out further, i.e. to flatten your overall pitch. Lower volume in the horn and neck = sharper. Greater volume in the horn and neck = flatter.
If your tone production technique doesn't change, and you want to play the same mouthpiece, you would need a horn that plays flatter for you. If you were going to experiment with necks, you should just get a few and try them. The one that plays better for you, with the mouthpiece further in, is likely to to be one with more volume, not less.
There is a lot more to it than this, but I'm out of time.
I've read your reply a couple of times... It all makes sense until the "so far out because you play sharp"... Have i not explained the problem well enough?
The issue is that it's playing incredibly flat where it sits and will not go in further into the neck as opposed to needing the piece to be further out on the cork to make it play flatter... which seems to be what you're referring to. Am I missing something ?