Changing tyre width does not affect brake geometry, If you have a brake hole in the seatstay bridge the critical dimension is the drop from centre of hole to the rim. You can get caliper brakes in std and deep drop.
Carerra CX often use Tektro Oryx cantilever brakes which are OK. Setup is critical, getting the mechanical advantage to suit the brake. Some systems use a "button" style cable, others a separate straddle cable. If using the later, you MUST fit a catching device in case of main cable breakage. A loose straddle cable will grip the tyre with immediate results. Use mudguards or a reflector mount to hold a loose straddle cable safely off the tyre.