You may have tried some of these, but here goes:

Have you tried different containers? Is there any chance he has a hard
time getting into the box? Is it covered or open? Big enough?

Have you tried putting any of his poops into the box to encourage him
to put more in there?

How many boxes do you have? Have you tried putting him in a room with
the box and no other cats (so that he can't possibly be attacked while
in the box)?

Which types of litter have you tried? Have you tried different
textures such as small grains, pellets, crystals, etc?

How about scents and types? Many cats cannot stand scented litter. And
some will like wheat, but not clay, or something else.

Has he shown any preference for anything? You mentioned that he goes
in the litter if it is on a flat surface. How about a very large,
shallow pan? You could try a film developing pan or something similar.
It would be wide and flat, but still hold litter.

You can also try deep litter and shallow litter. I had a hard time
with Chase til I discovered he wanted fine grain litter and extremely
deep litter. He likes to dig big time.

Do you know any of his history? How old is he? Was he an outdoor kitty
before?

I have no advice for the cover problem. My newest kitty, Jay Jay, does
not cover either. He actually paws at the litter after he is done, be
he doesn't bother to paw at the right spot.

Unfortunately, there are a ton of possible reasons why a cat would
object to the box.

Health reasons
box too small, or too hard to get to
location - high traffic, or easy to get attacked.
not enough boxes
    - some cats prefer to use one box for pee, and one for poop
scented litter
wrong texture litter
bad association with litter box
and more

We had an issue with a cat that was adopted from the shelter where the cat
would eat, and then back up a couple feet to go poop.  Our only conclusion
was that this was because the cat had been in a cage for a long time (it's a
long story about the cat, very special needs, would not come out of the cage
even if we left it open for him).  They figured he was used to having a
litterbox near where he ate.
Anyway, the adopters started putting a litterbox a couple feet away from the
food.  He began using it, and they slowly started moving the box farther and
farther away until they got it into an acceptable location.
I know it might be temporarily gross to have a litterbox in the kitchen or
where ever you feed your cats, but if it can help this cat use the litterbox
consistently, it will be worth it.  It sounds like you tried the opposite,
moving the food to the litterbox location, but that didn't work because of
the other cat.
Can you ask the rescue about his origins?   Was he a stray?  If he was,
maybe try some dirt from outside in the litterbox.  He might not recognize
that he is supposed to go in the litter if he was used to going in the great
outdoors.
Also see Meghan's post about using a low, shallow pan if that's what he
likes.

Good luck.

-Kelly