You could most likely fit 35" tires on it stock ride height but as soon as you ran over a pebble or sneeze anywhere near the steering wheel it would rub.
But in all seriousness, 35" tires on it now would fit but they will rub. I'd imagine somewhere around the 75% or more area and they'd start to rub. My 2001 2500 with a cummins I used to have had the 315's that are basically 35" and I had a 3" suspension lift on it, and in the front the tires were rubbing some piece of the body sticking out in the wheel well, so I just bent it out of the way with a hammer. Plenty of clearance otherwise, it was just that sticking out from the side. Granted a 3/4 ton truck sits higher from factory, but still, you should be able to make them work. For what it's worth though, I'd recommend 33" tires. I think you'd be fine either way because it won't get much use, but would probably be better for your application.
Great looking truck though. Nothing I like more than a 2nd gen ram in red.