I’ve never tried to smoke a whole Turkey in the 340-did a turkey breast once and had to turn it from side to side while cooking. There’s just not enough headroom. You might be able to spatchcock a small turkey and get it to fit, but even a large spatchcocked chicken takes up most of the room in my experience.
I did a 20 lb bird in 2 sessions last year. Went 1 step beyond spatchcock and split into 2 separate halves. Each half fit nicely over the drip tray. Cook time was 3:45 @ 275 F
Obviously, this is only an option if you don't need the entire bird done at the same time.