^^^ exactly. Butts, picnics, fresh hams, briskets, chuck roasts, etc. are much better after resting for 2 hours. That gives plenty of time to ramp those temps up to 350 for the chicken. Frankly, chicken is better done at higher temperatures that is normally considered best for those larger (thicker) cuts of meat.
Chicken and burgers go together well. I like my steaks medium rare, so the temps are more in the 500 range and really too hot for chicken.
Otherwise, you can cook chicken at 250-270. I wouldn't with the skin on and expect the skin to be worth a darn. In that 250-270 range most any cut of large meat (listed above) will do fine.
For example, with a 9lb butt in there, I'd smoke it at about 250-260 for say 4 hours, then foil it. In that temp range you're usually looking at about 10 hours total. At the 9th hour put the chicken on. 10th hour remove the foiled butt and place in a warmer of some kind (let it rest). 11th hour remove the chicken and start pulling the butt.
Something like that. However, frankly, I'd just add that 2nd hour to the end (like described at first).