It would appear to me that the problem lies actually in the mouthpiece itself. Sometimes the metal plug in the plastic slips out a fraction of an inch. You might just try putting a little pressure inwards on the mouthpiece while it's in the pipe and see if it'll slide in a little further. Or find something that is the same diameter as the outside of the metal insert to push against.
I have a meer that has an insert like this and it is a little loose. All I do is push against it and it'll slide in to where it looks good.