This question really stumped me. I spent a number of hours messing around with different redstone contraptions, trying to make something that matched what you wanted. I was able to get somewhat close but I wasn't left satisfied by it. Here is that contraption regardless:
Solution 1: The Pit
It starts with two hoppers transferring the contents of the trash chest into a chest in a minecart (Note: You should be able to make a two cart version with just a few adjustments). That minecart sits on a powered rail that is connected to "the junction". More on that later. The rail itself points towards a pit, 14 blocks deep. At the bottom sits one piece of obsidian, as seen here:
Now back to "the junction". "The junction" is what I named the three paths of redstone all connected to one lever. The right path leads to a redstone alternator in order to trigger the powered rail. The center path consists of 6 redstone repeaters all set to max time that terminates at a sticky piston. The left path is an alternator that leads to a simple redstone line in order to trigger the centerpiece of it all, a block of TNT.
In practice, you load up the center area with a block of TNT, as seen in the overview image, then place a minecart chest on the rail before pulling the lever. The cart will fall to the obsidian block, the TNT will be lit, and, after a small delay, the TNT will fall 16 blocks and blow up the cart, spilling its contents everywhere in order to despawn.
Note: As I write this now, I also realize you could put a obsidian block for the TNT to land on and you don't have to worry about timing at all which should let you compact it quite a bit more.
This was the closest I could get but I'm not really happy with it. It costs a minecart and chest (or two) every time along with a block of TNT. And while it's possible to save chests in order to reuse, I couldn't find a way to save the cart. So I went to find another solution.
My next solution is just as much as a bit of a hackjob. I ran out of ideas for some redstone contraption or clever chest usage and so I turned to different methods.
Solution 2: Overkill
I made a resource pack that removes the sound from the game.
This might not be ideal as the click is used for dispensers, buttons, menu buttons, levers, etc, but it undeniably makes it so you never have to sit through 6 minutes of clicking plus you don't even have to make some big machine with it.
You can find it here: QuietDroppers
I do hope someone finds a better solution though.
This is the circuit I had in mind:
It's actually pretty simple: Put one comparator on the hopper before your standard sorting system and one on the hopper after it. Then you combine them with an AND gate where one side is delayed and one side is inverted.
The idea: An item that goes through the hopper before the sorting system (which are all items that go through the system) activate the first comparator and with a delay deactivates the first redstone torch, the redstone next to it and activates the second redstone torch and the lamp. But when an item also goes through the hopper after the sorting system (which is everything except soul sand in my example), then it activates the second comparator and keeps the redstone line active, thereby keeping the second redstone torch and the lamp inactive. The timing is set up so that the first redstone lamp toggling and the repeater on the second comparator activating happen at almost the same time. There is one game tick delay because of hopper speed, but the redstone torch luckily ignores one tick off pulses.
This means that only soul sand going through the system will activate the lamp, which is exactly what you wanted. What happens inside the sorting system doesn't matter for this circuit, as long as it takes all soul sand items and nothing else.
Of course you also have to make sure that your sorter doesn't lose its only soul sand item, for that you just need to put 8 of the to be sorted item into the first slot instead of one:
This circuit relies on the speed of hopper lines. I assume your modded hoppers just process batches of 8 and aren't actually faster in sending items through hopper lines. If they are, you have to adjust the repeaters on the first comparator accordingly.
This circuit can also be put on the same side as the sorting circuit, for that you just need to replace the first piece of redstone dust after the second comparator with another repeater and increase the delay in the first comparator by one more tick. I didn't do it in the screenshot to make the circuit clearer.
Best Answer
The power levels of a comparator are spread evenly over the space in a chest. Since there are much more than 15 slots in a chest, you can't have three different comparator outputs from changing just one slot, not even with items of stack size 16 or 1.
What you could do is using a hopper, hopper minecart or furnace (you can put any item in, just be careful not to shift-click something smeltable and also one of the surprisingly many fuels).
The Minecraft wiki has a good table for the amount of items in different containers and their comparator power levels: https://minecraft.gamepedia.com/Redstone_Comparator archive