Make-up Air intake tripping smoke alarm
If the smoke is enough to trip the detector, but not enough to be objectionable to you, then you might consider replacing the smoke detector. A photoelectric type may be less prone to cooking-related nuisance alarms. A heat detector definitely will not trip for cooking smoke. A change to photoelectric is pretty easy. Check with your local authority to determine whether a heat detector would be acceptable for the location.
If the barbecue smoke comes from the fuel (ie wood or charcoal) I'd consider changing to a propane or natural gas fuel.
I don't love the idea of blocking the make-up air intake -- it would be all too easy for somebody to forget to unblock it after cooking, particularly if they don't understand the purpose of the intake. That said.. The grille covering the make up air intake may be steel. If it is then a magnetic sign could be an option for covering the intake. These are sometimes sold for covering heat registers inside the house as well. Ordinarily one would have to get within reach of the grille to place the cover, but if you secure it with rings or something at one end, then you could probably use a stick of some kind to lift it into place and to pull it back down again after cooking.