It probably is required by current codes, for what that's worth. CO detectors are required for new residences, even if there are no combustion appliances or fireplaces, at least in most areas.
If you're worried about 12V battery drain, I would suggest getting a battery-operated residential CO alarm or combination CO/smoke alarm. These also have the advantage of continuing to work if your 12V battery gets discharged for some reason, and not giving any false alarms when the 12V power is low.
If your BBQ is gas, it is certainly capable of producing a goodly amount of CO. In the open air that's not much of a problem because it has a chance to dissipate quickly, but if it gets blown into or through the TT it could reasonably collect and be unhealthy (but presumably delicious-smelling).