As of the latest, latest snapshot (14w02b), this is most certainly possible. In fact, there's an easier way to do the whole thing now, which is awesome. With the /blockdata command added in the snapshot, you can modify and overwrite the contents of a tile entity (including a chest), and much, much more. The applications of /blockdata go beyond what I can fully comprehend at this point in time, but, for this specific case, it's not too hard.
Let's say for example your equipment chest is supposed to contain a notch apple, a gold sword, a bow, and a stack of arrows. The command for that would be /blockdata X Y Z {Items:[{id:322,Damage:1,Count:1,Slot:0},{id:283,Count:1,Slot:1},{id:261,Count:1,Slot:2},{id:262,Count:64,Slot:3}]}, replacing X Y and Z as appropriate. There's a lot of stuff going on in that command, but essentially, you are changing the data of the chest to contain the specified items. It's the same concept as using /setblock to spawn a new chest with these items, except instead of replacing the chest, you're overwriting its data.
In this more specific case, you want /blockdata 264 4 997 {Items:[{id:274,Count:1,Slot:0},{id:275,Count:1,Slot:1}]}.
And, of course, if you want an empty chest, all you need to do is use /blockdata X Y Z {Items:[]}
This is due to the South-East rule. Despite a lack of mention in that page, this quirk does affect redstone. What's happening is that the dropper on top receives an item from the bottom dropper before it's instructed to fire when the comparator is on the north or the west sides of the droppers (opposite to what you said). When the comparators are on the east or south sides, the redstone power is cut before the top dropper is instructed to fire for the last item.
But it's more complicated than that. It's not just where the comparator is placed in relation to the dropper, but also where the rest of the redstone is placed. I've set up the same elevator, but mirrored, and it no longer emptied the top dropper.
My suggestion would be to either use a different item elevator, or find which of the 8 versions is going to work best for you.
Best Answer
It's because your top hopper is pointed directly into the hopper just below it. Point the hopper on the very top to the side, and it will work as you expect.
With the configuration you have, your second hopper from the top is getting 2 items per "cycle". (The top hopper is pushing one item / cycle into the 2nd hopper. The 2nd hopper is pulling one item / cycle from the top hopper.)
Now, the third hopper can only pull one item / cycle from the second hopper, so the second hopper pushes the "extra" item into the top chest.