Sorry I misunderstood with my earlier post. You have a large dial thermometer you're trying to fit into the door, not our normal probe thermometers like we stick in meat.
Usually dial thermometers are mounted on the front door, but a side wall works too so it's not moving and possibly hitting your meat. But you'll need to have access to read it on a side wall so that's why vendors don't put them there.
If you go with the door, make sure you don't mount it where it could hit a shelf, or a large piece of meat, but just below where a shelf would be.
If it's electric, it's probably double (but thin) wall construction...that gives better thermal insulation and electric smokers are generally starved for power so can't waste any with single walls.
7/8 hole sounds like it has a 1/2" National Pipe Thread boss on it. You'll need to drill a .875" (actually probably .840) in the outer wall and then you'll need an even larger hole on the inner wall so you can get a socket or wrench on the nut that tightens it to the outer thin wall. Can you exchange it for something with a smaller boss? Do you have a nut for it?