Two thoughts come to me - NO treated wood, NO imported plywood -----
I would NOT use treated wood.
You do not KNOW if it's ok or not, and in a smoking chamber, with minimum draft and the same smoke hanging around the meat inside that enclosure bothers me.
And understand, I am not afraid of treated wood of any type like some are.
Also, treated is not necessary just because it's outside.
Untreated wood can last for yearssss, Especially if not exposed to termites (Above ground a bit) or like exposed to constant moisture, in a hot VERY humid climate, in total shade.
I bet you can find hundreds of cobbled together OLD smokers in Kentucky or wherever countryside like.
If I used plywood, I would make sure it was USA made, not imported, as much is, especially at Big Box stores.
The chances of somekinda weird adhesive or chemicals used in it's manufacture, then "offgassing" around your meat is VERY possible if imported.
WAY less in USA products.
You can look into it, it's not just my opinion.
Sounds like a great project !! Enjoy !!