Our local distributors build up the tyres with inserts. Depends on what is available to you locally. You won't go far wrong with a pre-mount.
New carpet less grip? Not necessarily. As the track layout gets rubbered in, the grip gets higher; but as the carpet gets worn (over a period of years), the grip may get a little less.
If you haven't got enough grip, and the tyre rules are open, just go down a grade.