I wouldn't put a box in the basement! Now that he's used to having a box near where he sleeps, he's going to think the basement's awfully far away. I'd don't think he'd be willing to make that trip, especially if he had to go through a cat door to get there.
Do you have room in the bathroom to put two boxes side by side? If you don't, maybe there's another place on the main levels of your house where you can do that. If you can't do two boxes in the bathroom, the "safest" solution would be to buy two new boxes and put them side by side somewhere else. Since his litter box habits are already a bit shaky, you don't want to do anything to make things worse! When he's used to the new location, maybe you could take the box in the bathroom away.
Do you have a top on your box? If you do, try taking it off. He might want more room when he's having a bowel movement and feels confined in by the top on the box.
You can buy cat doors that will not allow heat loss and can be locked at night so the cat can't go out on the porch and come nose-to-nose through the screen with a bear. Here's a link for you: http://www.cat-dog-doors.com/
Much as I hate these things, I wonder if a self-feeder
would take the edge of his concern about the food dish running low. As long as the cannister was full, there should always be the same amount of food in the bowl. Just make sure your food fits through the opening before you put it down for your cat. For instance, NutroMax dry food seems to get stuck in self-feeders because the pieces are sort of big and an odd shape. If you do this, try to find a feeder that's stable and easy to eat out of. The one I linked to seems to work fairly well. You should be able to get it at your local PetsMart or PetCo store.
I'll be online tonight for couple more hours if you'd like to talk more.
I'm sending this as a request for information so you won't be prompted to pay for a second answer.