Yakiniku restaurants mostly smell like smoke to me. I'm not sure how discerning your vegan friend's nose is, so you've got a few options IMO:
First, your friend could just eat at a different restaurant the one time you go to Yakiniku. I bet if you are accommodating to their preference during other meals on the trip, it would be no problem for them to understand that for this one meal you'd all like to eat meat.
Or, they could sit at a different table. Yakiniku restaurants always have vegetable platters and other veggie dishes, and the grills are cleaned between guests, so if they were at their own table they could still enjoy their meal.
Or they could sit with everyone else and grill their veggie platter first for everyone to share, then leave once you guys get to the meat if the smell gets too strong for them. If you go during an "off" time (not during a popular time like dinner), there probably wouldn't be many other patrons, so perhaps they could stand it just for a short time.
Also, I think you could be a little less dismissive and a bit more understanding of this person's preferences. For whatever reason, they don't eat meat, and I don't think trying to convince them to eat it during the trip is going to do anything other than cause conflict in your group, which is no fun during travel! Accept that they wont eat meat, and understand that if you had dietary restrictions due to whatever reason, you would appreciate if people were understanding too!
Also, not everyone eats meat in Japan! How about finding your friend a shojin ryori restaurant during your stay? (
http://www.japan-guide.com/e/e3963.html) You can all enjoy the dishes prepared in vegetarian style, your vegan friend will be happy, and then if you do decide to go to yakiniku, perhaps your friend wont feel so left out!
Anyway, I hope you all enjoy your trip to Japan!