If I recall, the max temp on the MES is 275. You may not have the crispy skin but it should be able to handle it. I'm sure someone else who owns the MES and has done a turkey will be chiming in shortly. If you're worried about getting out of the danger temp zone quickly enough, you could spatchcock the bird which would lay it out flat resulting in a somewhat shorter cooking time.
While I don't have a MES, I have a cookshack, which is similar and only goes to 250 I have done full turkeys, turkey breasts, and boneless turkey breasts. It handled them all just fine. Like said above the skin won't be crispy but no on in my family likes the skin anyways. Just cook it like anything else to a proper internal temp and all will be good!
I have never done a large turkey before, but I have noticed poultry takes on a smoke flavor extremely easy. If I were you, I would smoke for an hour in the MES on the highest setting then put the bird in an oven at 300-350 to finish it. It will make the skin a lot better, wont take as long and will still have a smokey flavor. This is the way I do my chicken and it turns out great, where I had a problem using just the MES for the whole cooking process. Plus this would insure the turkey isnt in the danger zone for too long. Also, brine it! It's amaising to me how just a couple hours of brine can add so much to a bird.
What Reichi said..... Also I have one of those wire Turkey stands that you shove up the turkeys butt. It keeps the bird spread open so that the interior gets plenty of heat in the center. This speeds up the cooking and keeps it moist. In my opinion this is the safest way to do a Turkey.
If they made a kitchen rotisserie large enough to do a 20 pound Turkey like the Showtime Rotisserie I use for Chickens, I'd buy it in a heartbeat.