Sorry didn't get back to you sooner. Been under the weather since last night.
Okay, since it is a 22 lb frozen turkey it is going to take 4-5 days to thaw in the refrigerator. Could take up to a day longer depending how cold your refrigerator is. If you put it in a cooler with ice jugs to thaw in a cool garage, you're solidly in the 4-5 day range. Just change the ice jugs when they get too melted. I'm not a fan of thawing a turkey in cold water.
Brining. Butterball turkeys, even the all-natural ones, are injected with a self-basting brine and flavorings. You can brine it if you want, just cut the salt in the brine to 1/2 to 1/3rd of the amount the recipe calls for or the meat will be too salty. You can keep the sugar the same. A brined turkey is more tolerate of overshooting the temps. You'll still have a juicy bird. A simple brine for an 22 lb injected bird is (I already cut the salt below so no need to reduce it):
2 gallons of ice water.
1/2 cup kosher salt
1 cup brown sugar
8 smashed garlic toes
A bunch of fresh thyme, or 4 Tbs dried thyme.
Brine for 24 hours in a cooler, breast down. You can add more ice to the brine to keep it cold. Discard the brine when finished.
Do not stuff the bird.
On the day you are ready to smoke it, fire up the smoker to 350F-375F. Use a 50-50 mix of hickory (for flavor) and cherry (for color). There are a ton of ways you can prep the skin but to keep it simple I pat the skin relatively dry with a paper towel, leaving it just a little moist, then give it a decent coating of a your favorite BBQ rub.
Throw it in the smoker breast side up and after 3 hours use a meat thermometer to check the temperature of the breast and the thigh to see how it is doing. Aim for 165F in the breast and 175F in the thigh. Often the breast can climb to 185F before the thigh gets to 175F. That's where the brining comes in handy.
Have fun!