https://factoriobin.com/post/ket9xy

A recycling loop to sort the output, every splitters sorts an output, then goes to a priority output splitter to the provider chest, and any overflow goes to the next splitter... repeat until we have filtered everything, then any overflow gets recycled again.... abusing robots to move the sorted output to the rest of the base. Have 8 of these at the train depot to sort all incoming trains.

I originally used splitters but then switched to just filtering the inserters pulling from the loop. Since the loop recycles anyway it didn't matter if they missed some/the splitters felt like overkill.
For overflow I had inserters controlled by logi levels feeding into recyclers

After some amount of pain with sushi belts, I've just stolen the design from Nilaus with passthrough filtering splitters and bots, so very much similar to your setup but two main belts.

https://youtu.be/qr_Lep5UCJU?t=37m2s

Not at pc so don't have a BP but basically like this:

The outter belt is the overflow that goes through an array of speed boosted recyclers and the output goes back to the inner main as priority input.
You do have to account for the secondary outputs of the recycled overflow, so green circs, iron and copper plates, iron ore etc

Edit: the item filter of the storage chests must match the filter of the respective splitters (obviously), like this bots will prioritize these chests for both requests and "storage". Edit2: it is ready for quality upcycling but we didn't have time/felt like it at the moment

My main input:

Train comes in, dumps its load. That heads to main recyclers which sort into storage and excess is dumped onto quality line. Quality line is up north, next pic.

Line 1 is the next recycling loop I set up early. It was originally overflow for the main one, but then I converted it to quality. The * section is a "just in case" storage line. It holds another chest's worth of whatever of the main recycled materials.

Line 2 is the overflow from line 1 and also the overflow from main storage. There are requesters that only work over 10k of an item. This is also where I can dump custom items.

Line 3 is trash. It's intent is to void things, but I'll take any quality items I get.