What if you grill, using barbecue sauce? A conundrum indeed.
IMHO, in American English, barbecue is smoked and tends to involve barbecue sauce, grilling is grilling and sometimes involves barbecue sauce, and people who grill and call it barbecue are incorrect, because frankly there’s no comparison.
Admittedly, this was mostly lost on me the first half of my life, when I lived north of the Mason Dixon Line.